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PROGRESSION

Arithmetic Progression

In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers suchthat the difference of any two successive members of the sequence is a constant. For instance, the sequence3, 5, 7, 9, 11, 13, … is an arithmetic progression with common difference 2.

If the initial term of an arithmetic progression is a1 and the common difference of successive members is d ,then the nth term of the sequence is given by:

and in general

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

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

• 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:

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

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

An alternate form results from re-inserting the substitution: an = a1 + (n − 1)d :

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In 499 CE Aryabhata, a prominent mathematician-astronomer from the classical age of Indian mathematicsand Indian astronomy, gave this method in the  Aryabhatiya.

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

Product

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

where denotes the rising factorial and Γ denotes the Gamma function. (Note however that the formula isnot valid when a1 / d is a negative integer or zero.)

This is a generalization from the fact that the product of the progression is given by thefactorial n! and that the product

for  positive integers m and n is given by

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

Consider an AP

a,(a + d ),(a + 2d ),.................(a + (n − 1)d )Finding the product of first three terms

a(a + d )(a + 2d ) = (a2 + ad )(a + 2d ) = a3 + 3a2d + 2ad 2

this is of the form

an + nan − 1d n − 2 + (n − 1)an − 2d n − 1

so the product of n terms of an AP is:

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an + nan − 1d n − 2 + (n − 1)an − 2d n − 1 no solutions

Geometric ProgressionIn mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numberswhere each term after the first is found by multiplying the previous one by a fixed non-zero number calledthe common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. The sum of the terms of ageometric progression is known as a geometric series.

Thus, the general form of a geometric sequence is

and that of a geometric series is

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

Such a geometric sequence also follows the recursive relation

for every integer

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

The common ratio of a geometric series may be negative, resulting in an alternating sequence, with numbersswitching 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 andnegative infinity (due to the alternating sign).

Geometric sequences (with common ratio not equal to −1,1 or 0) show exponential growth or exponentialdecay, 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

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arithmetic progression yields a geometric progression, while taking the logarithm of each term in ageometric progression with a positive common ratio yields an arithmetic progression.

Geometric seriesMain article: Geometric series

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

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

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

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

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

For example:

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

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Then

For a series with only odd powers of r 

and

Infinite geometric seriesMain article: 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 ( | r | < 1 ). Its valuecan then be computed from the finite sum formulae

Since:

Then:

For a series containing only even powers of r ,

and for odd powers only,

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

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The formulae given above are valid only for | r | < 1. The latter formula is valid in every Banach algebra, aslong as the norm of r is less than one, and also in the field of   p- adic  numbers  if | r | p < 1. As in the case for afinite sum, we can differentiate to calculate formulae for related sums. For example,

This formula only works for | r | < 1 as well. From this, it follows that, for | r | < 1,

Also, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + · · · is an elementary example of a series that convergesabsolutely.

It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is

The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + · · · is a simple example of an alternating series that converges absolutely.

It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is

Complex numbers

The summation formula for geometric series remains valid even when the common ratio is a complexnumber . In this case the condition that the absolute value of r be less than 1 becomes that the modulus of r  be less than 1. It is possible to calculate the sums of some non-obvious geometric series. For example,consider the proposition

The proof of this comes from the fact that

which is a consequence of Euler's formula. Substituting this into the original series gives

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.

This is the difference of two geometric series, and so it is a straightforward application of the formula for infinite geometric series that completes the proof.

Product

The product of a geometric progression is the product of all terms. If all terms are positive, then it can bequickly computed by taking the geometric mean of the progression's first and last term, and raising thatmean to the power given by the number of terms. (This is very similar to the formula for the sum of terms ofan arithmetic sequence: take the arithmetic mean of the first and last term and multiply with the number of terms.)

(if a,r  > 0).

Proof:

Let the product be represented by P:

.

 Now, carrying out the multiplications, we conclude that

.

Applying the sum of arithmetic series, the expression will yield

.

.

We raise both sides to the second power:

.

Consequently

and

,

which concludes the proof.