HSC Arts 11thMaharashtra State Board
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# Geometric Progression (G. P.)

#### description

• Nth Term of Geometric Progression (G.P.) - T_n=ar^(n-1)
• General Term of a Geometric Progression (G.P.)
• Sum of First N Terms of a Geometric Progression (G.P.) - S_n=a(r^n-1)/(r-1)
• Infinite Geometric Progression (G.P.) and Its Sum - S∞=a1−r;|r|<1S∞=a1-r;|r|<1
• Geometric Mean (G.M.)

#### notes

Let us consider the following sequences:  2,4,8,16,...,
we have a_1 = 2 , a_2/a_1 = 2 , a_3/a_2 = 2, a_4/a_3 = 2 and so on.
In above sequence the constant ratio is 2.  Such sequences are called geometric sequence or geometric progression abbreviated as G.P.

1) General term of a G .P:
Let us consider a G.P. with first non-zero term ‘a’ and common ratio ‘r’. The second  term is obtained by multiplying a by r, thus a_2 = ar. Similarly, third term is obtained by multiplying a_2 by r. Thus, a_3 = a_2r = ar^2, and so on.
The nth term of a G.P. is given by a_n =ar^(n-1).
The series a + ar + ar^2 + ... + ar^(n–1)or a + ar + ar^2 + ...+ ar^(n–1) +... are called finite or infinite geometric series, respectively.

2)  Sum to n terms of a G .P.:
The first term of a G.P. be a and the common ratio be r.
s_n =(a(1-r^n))/1-r        or  s_n = (a(r^n -1))/r-1

3)  Geometric Mean (G .M.):
The geometric mean of two positive numbers a
and b is the number sqrt (ab) .
G_1, G_2,…, G_n be  n numbers between positive numbers a and b such that a,G_1,G_2,G_3,…,G_n, b is a G.P.

G_n =ar^n =a (b/a)^(n/(n + 1))

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Geometric Progression [00:07:06]
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