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From Wikipedia

Geometric progression

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. The sum of the terms of a geometric progression is known as a geometric series.

Thus, the general form of a geometric sequence is

a,\ ar,\ ar^2,\ ar^3,\ ar^4,\ \ldots

and that of a geometric series is

a + ar + ar^2 + ar^3 + ar^4 + \cdots

where r≠ 0 is the common ratio and a is a scale factor, equal to the sequence's start value.

Elementary properties

The n-th term of a geometric sequence with initial value a and common ratio r is given by

a_n = a\,r^{n-1}.

Such a geometric sequence also follows the recursive relation

a_n = r\,a_{n-1} for every integer n\geq 1.

Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio.

The common ratio of a geometric series may be negative, resulting in an alternating sequence, with numbers switching from positive to negative and back. For instance

1, −3, 9, −27, 81, −243, …

is a geometric sequence with common ratio −3.

The behaviour of a geometric sequence depends on the value of the common ratio.
If the common ratio is:

  • Positive, the terms will all be the same sign as the initial term.
  • Negative, the terms will alternate between positive and negative.
  • Greater than 1, there will be exponential growth towards positive infinity.
  • 1, the progression is a constant sequence.
  • Between −1 and 1 but not zero, there will be exponential decay towards zero.
  • −1, the progression is an alternating sequence (see alternating series)
  • Less than −1, for the absolute values there is exponential growth towards positive and negative infinity (due to the alternating sign).

Geometric sequences (with common ratio not equal to −1,1 or 0) show exponential growth or exponential decay, as opposed to the Linear growth (or decline) of an arithmetic progression such as 4, 15, 26, 37, 48, … (with common difference 11). This result was taken by T.R. Malthus as the mathematical foundation of his Principle of Population. Note that the two kinds of progression are related: exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression with a positive common ratio yields an arithmetic progression.

Geometric series

A geometric series is the sum of the numbers in a geometric progression:

\sum_{k=0}^{n} ar^k = ar^0+ar^1+ar^2+ar^3+\cdots+ar^n. \,

We can find a simpler formula for this sum by multiplying both sides of the above equation by 1 − r, and we'll see that

\begin{align}

(1-r) \sum_{k=0}^{n} ar^k & = (1-r)(ar^0 + ar^1+ar^2+ar^3+\cdots+ar^n) \\ & = ar^0 + ar^1+ar^2+ar^3+\cdots+ar^n \\ & {\color{White}{} = ar^0} - ar^1-ar^2-ar^3-\cdots-ar^n - ar^{n+1} \\ & = a - ar^{n+1} \end{align}

since all the other terms cancel. Rearranging (for r≠ 1) gives the convenient formula for a geometric series:

\sum_{k=0}^{n} ar^k = \frac{a(1-r^{n+1})}{1-r}.

If one were to begin the sum not from 0, but from a higher term, say m, then

\sum_{k=m}^n ar^k=\frac{a(r^m-r^{n+1})}{1-r}.

Differentiating this formula with respect to r allows us to arrive at formulae for sums of the form

\sum_{k=0}^n k^s r^k.

For example:

\frac{d}{dr}\sum_{k=0}^nr^k = \sum_{k=1}^n kr^{k-1}=

\frac{1-r^{n+1}}{(1-r)^2}-\frac{(n+1)r^n}{1-r}.

For a geometric series containing only even powers of r multiply by 1-r^2:

(1-r^2) \sum_{k=0}^{n} ar^{2k} = a-ar^{2n+2}.

Then

\sum_{k=0}^{n} ar^{2k} = \frac{a(1-r^{2n+2})}{1-r^2}.

For a series with only odd powers of r

(1-r^2) \sum_{k=0}^{n} ar^{2k+1} = ar-ar^{2n+3}

and

\sum_{k=0}^{n} ar^{2k+1} = \frac{ar(1-r^{2n+2})}{1-r^2}.

Infinite geometric series

An infinite geometric series is an infinite series whose successive terms have a common ratio. Such a series converges if and only if the absolute value of the common ratio is less than one (&nbsp;|&nbsp;r&nbsp;|&nbsp;<&nbsp;1&nbsp;). Its value can then be computed from the finite sum formulae

\sum_{k=0}^\infty ar^k = \lim_{n\to\infty}{\sum_{k=0}^{n} ar^k} = \lim_{n\to\infty}\frac{a(1-r^{n+1})}{1-r}= \lim_{n\to\infty}\frac{a}{1-r} - \lim_{n\to\infty}{\frac{ar^{n+1}}{1-r}}

Since:

r^{n+1} \to 0 \mbox{ as } n \to \infty \mbox{ when } |r| < 1.

Then:

\sum_{k=0}^\infty ar^k = \frac{a}{1-r} - 0 = \frac{a}{1-r}

For a series containing only even powers of r,

\sum_{k=0}^\infty ar^{2k} = \frac{a}{1-r^2}

and for odd powers only,

\sum_{k=0}^\infty ar^{2k+1} = \frac{ar}{1-r^2}

In cases where the sum does not start at k = 0,

\sum_{k=m}^\infty ar^k=\frac{ar^m}{1-r}

The formulae given above are valid only for |&nbsp;r&nbsp;|&nbsp;<&nbsp;1. The latter formula is valid in every Banach algebra, as long as the norm of r is less than one, and also in the field of p-adic numbers if |&nbsp;r&nbsp;|p&nbsp;<&nbsp;1. As in the case for a finite sum, we can differentiate to calculate

Arithmetic progression

In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For instance, the sequence 3, 5, 7, 9, 11, 13, &hellip; is an arithmetic progression with common difference 2.

If the initial term of an arithmetic progression is a_1 and the common difference of successive members is d, then the nth term of the sequence is given by:

\ a_n = a_1 + (n - 1)d,

and in general

\ a_n = a_m + (n - m)d.

A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression.

The behavior of the arithmetic progression depends on the common difference d. If the common difference is:

  • Positive, the members (terms) will grow towards positive infinity.
  • Negative, the members (terms) will grow towards negative infinity.

Sum

The sum of the members of a finite arithmetic progression is called an arithmetic series.

Expressing the arithmetic series in two different ways:

S_n=a_1+(a_1+d)+(a_1+2d)+\cdots+(a_1+(n-2)d)+(a_1+(n-1)d)
S_n=(a_n-(n-1)d)+(a_n-(n-2)d)+\cdots+(a_n-2d)+(a_n-d)+a_n.

Adding both sides of the two equations, all terms involving d cancel:

\ 2S_n=n(a_1+a_n).

Dividing both sides by 2 produces a common form of the equation:

S_n=\frac{n}{2}( a_1 + a_n).

An alternate form results from re-inserting the substitution: a_n = a_1 + (n-1)d:

S_n=\frac{n}{2}[ 2a_1 + (n-1)d].

In 499 CE Aryabhata, a prominent mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, gave this method in the Aryabhatiya(section 2.18) .

So, for example, the sum of the terms of the arithmetic progression given by an = 3 + (n-1)(5) up to the 50th term is

S_{50} = \frac{50}{2}[2(3) + (49)(5)] = 6,275.

Product

The product of the members of a finite arithmetic progression with an initial element a1, common differences d, and n elements in total is determined in a closed expression

a_1a_2\cdots a_n = d^n {\left(\frac{a_1}{d}\right)}^{\overline{n}} = d^n \frac{\Gamma \left(a_1/d + n\right) }{\Gamma \left( a_1 / d \right) },

where x^{\overline{n}} denotes the rising factorial and \Gamma denotes the Gamma function. (Note however that the formula is not valid when a_1/d is a negative integer or zero.)

This is a generalization from the fact that the product of the progression 1 \times 2 \times \cdots \times n is given by the factorial n! and that the product

m \times (m+1) \times (m+2) \times \cdots \times (n-2) \times (n-1) \times n \,\!

for positive integers m and n is given by

\frac{n!}{(m-1)!}.

Taking the example from above, the product of the terms of the arithmetic progression given by an = 3 + (n-1)(5) up to the 50th term is

P_{50} = 5^{50} \cdot \frac{\Gamma \left(3/5 + 50\right) }{\Gamma \left( 3 / 5 \right) } \approx 3.78438 \times 10^{98}

Consider an AP

a,(a+d),(a+2d),.................(a+(n-1)d)

Finding the product of first three terms

a(a+d)(a+2d) =(a^{2}+ad)(a+2d) =a^{3}+3a^{2}d+2ad^{2}

this is of the form

a^{n} + na^{n-1} d^{n-2} + (n-1)a^{n-2}d^{n-1}

so the product of n terms of an AP is:

a^{n} + na^{n-1} d^{n-2} + (n-1)a^{n-2}d^{n-1} no solutions


Geometric mean

The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, which is what most people think of with the word "average", except that the numbers are multiplied and then the nth root (where n is the count of numbers in the set) of the resulting product is taken.

For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product; that is '. As another example, the geometric mean of three numbers 1, ½, ¼ is thecube root of their product (1/8), which is 1/2; that is '.

The geometric mean can also be understood in terms of geometry. The geometric mean of two numbers, a and b, is the length of one side of a square whose area is equal to the area of a rectangle with sides of lengths a and b. Similarly, the geometric mean of three numbers, a, b, and c, is the length of one side of a cube whose volume is the same as that of a right cuboid with sides whose lengths are equal to the three given numbers.

The geometric mean only applies to positive numbers. It is also often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as data on the growth of the human population or interest rates of a financial investment. The geometric mean is also one of the three classic Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean.

Calculation

The geometric mean of a data set \{a_1,a_2 , \ldots,a_n\} is given by:

\bigg(\prod_{i=1}^n a_i \bigg)^{1/n} = \sqrt[n]{a_1 a_2 \cdots a_n}.

The geometric mean of a data set is less than or equal to the data set's arithmetic mean (the two means are equal if and only if all members of the data set are equal). This allows the definition of the arithmetic-geometric mean, a mixture of the two which always lies in between.

The geometric mean is also the arithmetic-harmonic mean in the sense that if two sequences (an) and (hn) are defined:

a_{n+1} = \frac{a_n + h_n}{2}, \quad a_0=x

and

h_{n+1} = \frac{2}{\frac{1}{a_n} + \frac{1}{h_n}}, \quad h_0=y

then an and hn will converge to the geometric mean of x and y.

This can be seen easily from the fact that the sequences do converge to a common limit (which can be shown by Bolzano-Weierstrass theorem) and the fact that geometric mean is preserved:

\sqrt{a_ih_i}=\sqrt{\frac{a_i+h_i}{\frac{a_i+h_i}{h_ia_i}}}=\sqrt{\frac{a_i+h_i}{\frac{1}{a_i}+\frac{1}{h_i}}}=\sqrt{a_{i+1}h_{i+1}}

Replacing arithmetic and harmonic mean by a pair of generalized means of opposite, finite exponents yields the same result.

Relationship with arithmetic mean of logarithms

By using logarithmic identities to transform the formula, we can express the multiplications as a sum and the power as a multiplication.

\bigg(\prod_{i=1}^na_i \bigg)^{1/n} = \exp\left[\frac1n\sum_{i=1}^n\ln a_i\right]

This is sometimes called the log-average. It is simply computing the arithmetic mean of the logarithm transformed values of a_i (i.e., the arithmetic mean on the log scale) and then using the exponentiation to return the computation to the original scale, i.e., it is the generalised f-mean with f(x) = log&nbsp;x.

For normal frequency:

GM = Antilog\left[\frac1n\sum\ logx\right]

For frequency distribution:

GM = Antilog\left[\frac1n\sum\ f logx\right]

Applications

Proportional growth

The geometric mean is more appropriate than the arithmetic mean for describing proportional growth, both exponential growth (constant proportional growth) and varying growth; in business this is known as the compound annual growth rate (CAGR). The geometric mean of growth over periods yields the equivalent constant growth rate that would yield the same final amount.

Suppose an orange tree yields 100 oranges one year and then 180, 210 and 300 the following years, so the growth is 80%, 16.7% and 42.9% for each year respectively. Using the arithmetic mean calculates a (linear) average growth of 46.5% (80% + 16.7% + 42.9% divided by 3). However, if we start with 100 oranges and let it grow 46.5% each year, the result is 314 oranges, not 300, so the linear average over-states the year-on-year growth.

Instead, we can use the geometric mean. Growing with 80% corresponds to multiplying with 1.80, so we take the geometric mean of 1.80, 1.167 and 1.429, i.e. \sqrt[3]{1.80 \times 1.167 \times 1.429} = 1.443, thus the "average" growth per year is 44.3%. If we start with 100 oranges and let the number grow with 44.3% each year, the result is 300 oranges.

Aspect ratios

The geometric mean has been used in choosing a compromise aspect ratio in film and video: given two aspect ratios, the geometric mean of them provides a compromise between them, distorting or cropping both in some sense equally. Concretely, two equal area rectangles (with the same center and parallel sides) of different aspect ratios intersect in a rectangle whose aspect ratio is the geometric mean, and their hull (smallest rectangle which contains both of them) likewise has aspect ratio their geometric mean.

In arithmetic and algebra, the cube of a number n is its third power&mdash; the result of the number multiplying by itself three times:

n3 = n&times; n&times; n.

This is also the volume formula for a geometric cube with sides of length n, giving rise to the name. The inverse operation of finding a number whose cube is n is called extracting the cube root of n. It determines the side of the cube of a given volume. It is also n raised to the one-third power.

A perfect cube (also called a cube number, or sometimes just a cube) is a number which is the cube of an integer.

The sequence of non-negative perfect cubes starts :

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 64000, 68921, 74088, 79507, 85184, 91125, 97736, 103823, 110592, 117649, 125000, 132651, 140608, 148877, 157464, 166375, 175616, 185193, 195112, 205379, 216000, 226981, 238328...

Geometrically speaking, a positive number m is a perfect cube if and only if one can arrange m solid unit cubes into a larger, solid cube. For example, 27 small cubes can be arranged into one larger one with the appearance of a Rubik's Cube, since 3&nbsp;&times;&nbsp;3&nbsp;&times;&nbsp;3 = 27.

The pattern between every perfect cube from negative infinity to positive infinity is as follows,

n3 = (n&minus; 1)3 + (3n&minus; 3)n + 1.

Cubes in number theory

There is no smallest perfect cube, since negative integers are included. For example, (&minus;4)&nbsp;&times;&nbsp;(&minus;4)&nbsp;&times;&nbsp;(&minus;4)&nbsp;=&nbsp;&minus;64. For any n, (&minus;n)3 = &minus;(n3).

Base ten

Unlike perfect squares, perfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 25, 75 and 00 can be the last two digits, any pair of digits with the last digit odd can be a perfect cube. With even cubes, there is considerable restriction, for only 00, o2, e4, o6 and e8 can be the last two digits of a perfect cube (where o stands for any odd digit and e for any even digit). Some cube numbers are also square numbers, for example 64 is a square number (8&nbsp;&times;&nbsp;8) and a cube number (4&nbsp;&times;&nbsp;4&nbsp;&times;&nbsp;4); this happens if and only if the number is a perfect sixth power.

It is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have digital root1, 8 or 9. Moreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by 3:

  • If the number is divisible by 3, its cube has digital root 9;
  • If it has a remainder of 1 when divided by 3, its cube has digital root 1;
  • If it has a remainder of 2 when divided by 3, its cube has digital root 8.

Waring's problem for cubes

Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes:

23 = 23 + 23 + 13 + 13 + 13 + 13 + 13 + 13 + 13.

Fermat's last theorem for cubes

The equation x3 + y3 = z3 has no non-trivial (i.e. xyz&ne; 0) solutions in integers. In fact, it has none in Eisenstein integers.

Both of these statements are also true for the equation x3 + y3 = 3z3.

Sums of rational cubes

Every positive rational number is the sum of three positive rational cubes, and there are rationals that are not the sum of two rational cubes.

Sum of first ''n'' cubes

The sum of the first n cubes is the nthtriangle number squared:

1^3+2^3+\dots+n^3 = (1+2+\dots+n)^2=\left(\frac{n(n+1)}{2}\right)^2.

For example, the sum of the first 5 cubes is the square of the 5th triangular number,

1^3+2^3+3^3+4^3+5^3 = 15^2 \,

A similar result can be given for the sum of the first yodd cubes,

1^3+3^3+\dots+(2y-1)^3 = (xy)^2

but {x,y} must satisfy the negative Pell equation x^2-2y^2 = -1. For example, for y = 5 and 29, then,

1^3+3^3+\dots+9^3 = (7*5)^2 \,
1^3+3^3+\dots+57^3 = (41*29)^2

and so on. Also, every evenperfect number, except the first one, is the sum of the first 2(p−1)/2odd cubes,

28 = 2^2(2^3-1) = 1^3+3^3
496 = 2^4(2^5-1) = 1^3+3^3+5^3+7^3
8128 = 2^6(2^7-1) = 1^3+3^3+5^3+7^3+9^3+11^3+13^3+15^3

Sum of cubes in arithmetic progression

There are examples of cubes in arithmetic progression whose sum is a cube,

3^3+4^3+5^3 = 6^3
11^3+12^3+13^3+14^3 = 20^3
31^3+33^3+35^3+37^3+39^3+41^3 = 66^3

with the first one also known as Plato's number. The formula F for finding the sum of an n number of cubes in arithmetic progression with common difference d and initial cube a3,

F(d,a,n) = a^3+(a+d)^3+(a+2d)^3+...+(a+dn-d)^3

is given by,

F(d,a,n) = (n/4)(2a-d+dn)(2a^2-2ad+2adn-d^2n+d^2n^2)

A parametric solution to,

F(d,a,n) = y^3

is known for the special case of d = 1, or consecutive cubes, but only sporadic solutions are known for integer d> 1, such as d = {2,3,5,7,11,13,37,39}, etc.

History

Determination of the Cube of large numbers was very common in many ancient civilizations. From Yahoo Answers

Question:I've never fully learned this concept, so I need help on some problems. I'm going post 3, but you don't have to answer all of the them. But if you do, thats obviously going to be an easy 10 points. I appreciate good explanations! Thank you very much for even just one answer to problem! (Remember, you are encouraging growth in the knowledge of the future human race. Thank you for ridding this world of ignorant idiots) 1) If x, 2x + 2, 3x + 3 are a geometric progression, what is the 4th term ? 2)For what values of x does the infinite geometric series converge? x2/3 x4/6 + x6/12 - (What does converge mean?) 3)The first three terms of an arithmetic sequence, in order, are 2x + 4, 5x 4, and 3x + 4. What is the sum of the first ten terms of the sequence? Hopefully, this will be the last time I ask total strangers for help, but I'm really quite desperate. I greatly appreciate any answer to any problem!!! Thanks so much! I understand #2 now, so you shouldn't do the problem because I already figured it out. I'm not level 2, and I can't rate, but I'll choose best answer.

Answers:Hi, 1) If x, 2x + 2, 3x + 3 are a geometric progression, what is the 4th term ? ..x...... .2x+2 -------..=.-------- 2x+2.....3x+3 x(3x+3) = (2x+2) 3x + 3x = 4x + 8x + 4 0 = x + 5x + 4 0 = (x + 4)(x + 1) x = -4 or x = -1 If x = -4, then the terms are -4, -6, and -9. The r value is 1.5, so the 4th term would be -13.5. If x = -1, then the terms are -1, 0, and 0. The r value is 0, so the 4th term would also be 0. Since that's a rather silly sequence, I'd go with -13.5 as the 4th term you wanted. 2)For what values of x does the infinite geometric series converge? x2/3 x4/6 + x6/12 - (What does converge mean?) I don't understand if those are fractional exponents or an exponent and fraction, etc. Can you clarify? Converge means it approaches a certain number. 3)The first three terms of an arithmetic sequence, in order, are 2x + 4, 5x 4, and 3x + 4. What is the sum of the first ten terms of the sequence? (5x - 4) - (2x + 4) = 3x - 8 (3x + 4) - (5x - 4) = -2x + 8 Since these differences must be the same, then 3x - 8 = -2x + 8 5x = 16 x = 16/5 The numbers are 10.4, 12, and 13.6. Their common difference is 1.6, so a = 10.4 and d = 1.6. S(10) = n/2(2a + d(n - 1)) S(10) = 10/2(2*10.4 + 1.6(10 - 1)) = 176 <==ANSWER I hope that helps with #1 and #3. Please clarify #2. :-)

Question:The measures of the interior angles of a convex polygon are in arithmetic progression. If the common difference is 5 degrees and the largest angle is 160 degrees, find the value of n. Show all work, enjoy!

Answers:The angles are going to be 160, 155, 150, 145, etc. The formula for the sum of the interior angles of an n-sided polygon is: (n - 2) * 180. So you need to keep adding terms until you get a multiple of 180: This happens when you get down to 120. 120 + 125 + 130 + 135 + 140 + 145 + 150 + 155 + 160 = 1260 = 180 * 7. So that means the polygon has 9 sides.

Question:Some people believe they can make money from a chain letter (they are usually disappointed). A chain letter works roughly like this: A letter arrives with a list of four names attached and instructions to mail a copy to four more friends and to send 1 to the top name on the list. When you mail the four letters, you remove the top name (to whom the money was sent) and add your own name to the bottom of the list. a) If no one breaks the chain, how much money do you receive? b) Let d sub n be the number of dollars you receive if there are n names on the list instead of 4, but you still mailed to four friends. Find a formula for d sub n. Please help me with this confusing problem. Also show your work and explain how you got to the answer. Thank you so much!

Answers:Need help with arithmetic/geometric series!!!!!!!!? Some people believe they can make money from a chain letter (they are usually disappointed). A chain letter works roughly like this: A letter arrives with a list of four names attached and instructions to mail a copy to four more friends and to send 1 to the top name on the list. When you mail the four letters, you remove the top name (to whom the money was sent) and add your own name to the bottom of the list. Johnny a) If no one breaks the chain, how much money do you receive ===================================================================== Diagram: http://www.flickr.com/photos/27678773@N06/4591444508/ Amount received = 31 money. Note: If $1 is charge then I received $31 on the 3rd chain leter. ans ====================================================================== b) Let d sub n be the number of dollars you receive if there are n names on the list instead of 4, but you still mailed to four friends. Find a formula for d sub n. Note: m = number of chain letter sent after the second letter (see diagram) ================================== d = (n-1)+(n)(n-1)+ m(n x n) Ans ================================== hope this helps Remember that Jesus loves you. Know Him in His words the Bible. God Bless Lim E

Question:I really don't understand arithmetic & geometric functional and recursive equations please help, i understand what the difference is but i mean, what is the difference between funtional and recursive and what are their basic equations, such a f(n) = 71+(n-1)1.03

Answers:A sequence of numbers is a list of numbers (finite or infinite). There are several ways you can describe the numbers (not using just words): (1a) a finite list can be written completely, (1b) an infinite list can be shown using the first several numbers, (2) a Functional formula is a formula for number a(n) using the number "n", (3) a Recursive formula for number a(n) uses the previous number (or numbers) a(n-1). Example. Even numbers starting from 2. (1a) Even numbers 2 to 20 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 (1b) 2, 4, 6, 8, 10, 12, ... (2) a(n) = 2*n+2 ... n = 0, a(0) = 2*0+2 = 2 ... n = 1, a(1) = 2*1+2 = 4 ... n = 2, a(2) = 2*2+2 = 6 ... n = 3, a(3) = 2*3+2= 8 (3) a(0) = 2, and a(n) = a(n-1) + 2 ... n = 0, a(0) = 2 ... n = 1, a(1) = 2+2 = 4 ... n = 2, a(2) = 4+2 = 6 ... n = 3, a(3) = 6+2= 8 This example is an arithmetic sequence. A geometric sequence would have the terms multiplied by some constant.

From Youtube

Sum of N Natural Numbers or Positive Integers Formula :SkyingBlogger.Com - Sum of n natural numbers or Sum of n positive integers is discussed in this videos. We derive a simple formula for sum of n natural numbers. By using the sum of n natural numbers we can easily calculate the sum of any numbers of natural numbers or positive integers. its really simple. Sum of n natural numbers positive integers = n(n-1) / 2 This sum of n natural numbers or positive integers is an special case of the formula - Sum of n numbers of an Arithmetic Sequence or Arithmetic Progression .