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Introduction

Progression

In mathematics:

y Arithmetic progression, sequence of numbers such that the

difference of any two successive members of the sequenceis a constant

y Geometric progression, sequence of numbers such that thequotient of any two successive members of the sequence isa constant


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ARITHMETIC PROGRESSIONS

If th

initial termof an arithmetic pro

ression is a1 and the common

difference of successive mem

ers is d

then the nth termof the sequenceis given by

and in general

finite portionof an arithmetic progression is called a finite arithmetic

progression and sometimes just called an arithmetic progression.

The behavior of the arithmetic progressiondepends on the common

difference d. If the commondifference is:

y Positive, the term will grow towards positive infinity

y Negative, the term will grow towards negative infinity.

The sumof the members of a finite arithmetic progression is called an

arithmetic series.

Expressing the arithmetic series in twodifferent ways

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

Dividing both sides by 2 produces a commonformof the equation:

n alternate form results from re-inserting the substitution: an = a1 + (n

1)d:


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In 499 CE

ryabhata, a prominent mathematic astronomer from theclassical age of Indianmathematic and Indian astronomy gave this

method in the

ryabhatiya.

So, for example, the sumof the terms of the arithmetic progression given

by an = 3 + (n-1)(5) up to the 50th term is

The product of the members of a finite arithmetic progression with aninitial element a1, commondifferences d, andn elements in total is

determined in a closed expression

where denotes the rising factorial andnotes the

ammafunction.However that the formula is not valid when a1 / d is a negative

integer or zero.

This is a generalizationfrom the fact that the product of the progressionis given by the factorial n! and that the product

For positive integers m andn 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


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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 + nan1dn

2 + (n 1)an 2dn

1

so the product of n terms of an AP is:

an + nan1dn

2 + (n 1)an 2dn

1 no solutions


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GEOMETRIC PROGRESSIONS

Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ... which converges to 2.

Example, the sequence 2, 6, 18, 54, ... is a geometric progression withcommon ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with

common ratio 1/2. The sumof the terms of a geometric progression is

known as a geometric series.

Thus, the general formof 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.


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A geometric series is the sumof the numbers in a geometric progression:

We canfind a simpler formula for this sum bymultiplying both sides ofthe above equation by 1 r, and we'll see that

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

Ifone were to begin the sumnot from 0, but from a higher term, saym,

then

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

For example:

For a geometric series containing only even powers of r multiply by 1 r2:


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Then

For a series withonlyodd powers of r

and

An infinite geometric series is an infinite series whose successive terms

have a common ratio. Such a series converges if andonly if the absolute

value the common ratio is less thanone ( | r | < 1 ). Its value can then be

computedfrom the finite sumformulae

Since:

Then:

For a series containing only even powers of r,

andfor odd powers only,


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In cases where the sumdoes not start at k = 0,

The formulae given above are validonlyfor | r | < 1. The latter formula isvalid in every Banach algebra, as long as the normof r is less thanone,

and also in the fieldof p-adic numbers if | r |p < 1.As in the case for afinite sum, we candifferentiate 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,infinite series /2 + 1/4 + 1/8 + 1/16 + is an elementary example of aseries 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

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 a geometric series whose first term is 1/2 and whose common ratio is

1/2, so its sum is


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The product of a geometric progression is the product of all terms If all

terms are positive, then it can be quickly computed by taking the

geometric mean of the progression's first and last term, and raising that

mean to the power given by the number of terms

(if a,r > 0)

Let the product be represented by P:

Now, carrying out the multiplications, we conclude that

Applying the sum of arithematic series, the expression will yield

We raise both sides to the second power:

Consequently

and

,

which concludes the proof


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