The geometrical progression is the succesion of term so that every term bears the fixed ratio to proceding term which is obtainable by doing multiplication to the precediing term by fixed quantity and the fixed ration is known as common ratio of the geometric progression.
OR
In modular arithmetic, geometric progression are studied under sequences and series. Let's few basics about the arithmetic progressions and Geometric Progression .
Terms of series a, ar, ar2, ar3,.... here every term bears constant ratio for the preceding term is called as geometric progression and constant ratio is known as common ratio.
Geometric Progression Examples
Example 1: 3 + 1 + $\frac{1}{3}$ + $\frac{1}{3^{2}}$ + $\frac{1}{3^{3}}$ + ..........
Geometrical series with common ratio $\frac{1}{3}$
Example 2: 2 - 2$\sqrt{3}$ + 6 - 6$\sqrt{3}$ + 18......
similar to arithmetic progressions the Geometrical series with common ratio -$\sqrt{3}$.
⇒ From the two examples it is seen that the signs of the terms of a Geometric Progressions must either be all alike or alternatively positive and negative.
The numbers in continued proportion are in Geometric Progressions, i.e.,
$\frac{x}{y}$ = $\frac{y}{z}$ = $\frac{z}{w}$ = .... = $\frac{1}{r}$ then
y = rz , z = ry, w = rz..
$\rightarrow$ y = xr, z = xr2, w = xr3
Nth Term Formula
If first term is a, common ratio is r and number of terms is n then t1 = a, t2 = ar, t3 = ar2,..... and so on.
We observe that the index of r on the right hand side is one less than the suffix of t on the left hand side in each of the equalities. Hence tn = arn-1 is the general term of the given geometric progressions.
If the product of three numbers in Geometric Progressions is given, take the terms as $\frac{a}{r}$, a, ar. But if the product of the numbers is not given, the terms are in the ordinary form.
Sum of Geometric Progression
Let a = First term, r = common ratio,
n = number of terms.
tn = nth term = arn-1, Sn = sum to n terms
Sn = a + ar + ar2 + ....+ arn-1 ....(i)
Multiply both sides of (i) by r, the common ratio.
rSn = ar + ar2 + ....+ arn-1 + arn
Subtracting (ii) from (i), we get
Sn(1 - r) = a - arn = a(1 - rn)
Sn = a$\frac{(1 - r^{n})}{1 - r}$
When r < 1, Sn = a$\frac{(1 - r^{n})}{1 - r}$
When r > 1, Sn = a$\frac{(r^{n} - 1)}{r - 1}$
Infinite Sum
Consider the Geometric Progressions a, ar, ar2...
When r < 1 we have Sn = a$\frac{(1 - r^{n})}{1 - r}$ = $\frac{a}{1 - r}$ - $\frac{ar^{n}}{1 - r}$
As n increases rn decreases and as n $\to$ $\infty$, rn $\to$ 0.
$\therefore$ S$\infty$ = $\frac{a}{1 -r}$ - $\frac{0}{1 - r}$ = $\frac{a}{1 -r}$
Sum to infinity exists only when r is numerically less than 1. i.e. |r|<1
Sigma Notation Series
The Greek letter σ (read as sigma) denotes the sum. When written before the nth term of series, it implies the sum of all terms obtained by giving to n the different values 1, 2, 3…n. Thus,
⇒ n stands for 1 + 2 + 3 + 4 + ...........+ n
⇒ n2 stands for 12 + 22 + 32 + 42 + ...........+ n2
⇒ n3 stands for 13 + 23 + 33 + 43 + ...........+ n3
⇒ stands for 1 + 1 + 1 + 1 + ...........+ 1 = n
⇒ 10 stands for 10 + 10 + 10 + 10 + ...........+10 = 10n
Special Series
1) Sum of first n natural numbers = $\frac{n(n + 1)}{2}$
2) Sum of squares of first n natural numbers = $\frac{n(n+1)(2n+1)}{6}$
3) Sum of cubes of first n natural numbers = $(\frac{n(n + 1)}{2})^{2}$