Write the formula for finding the sum of n terms of a G.P.

1.	Write the formula for finding the sum of n terms of a G.P.
Answer» Let ‘a’ be the first term and ’r’(≠) be the common ratio of the given G.P.The general teams of G.P. If S_n denotes the sum of ‘n’ terms, then, S_n = a((1 - rⁿ)/(1 - r)) r ≠ 1 This can also written as S_n = a((rⁿ - 1)/(r - 1)) Note: (1) S_n = a + ar + ar²+….. +ar^n-1 (2) If r = 1, then Sn = a + a + a +……………+a – na (3) If r = 1, then formulas for S_n fail (4) If r < 1, then use S_n = a((1 - rⁿ)/(1 - r)) (5) If r > 1, use S_n = a((rⁿ - 1)/(r - 1)) (6) Sum of an infinite G.P.,a + ar + ar² + ..... is a/(1 - r), - 1 r < 1 (7) If a, b, c are in G.P.then ‘b’ is the G.M. (geometric mean) between a and c, then b² = ac ⇒ b = √ac (8) If A and G are A.M. and G.M. of two given distinct positive real numbers, then relation between A and G is A > G. Sum of G.P. up to n terms if 1^st term is a and the common ratio is r. Sn =a.(1-rⁿ)/(1-r)………….(1) while r < 1 . or Sn =a.(r^n-1)/(r - 1)………(2) while r > 1 . Sum up to infinite terms = a/(1-r).

Answer»

Let ‘a’ be the first term and ’r’(≠) be the common ratio of the given G.P.The general teams of G.P. If S_n denotes the sum of ‘n’ terms, then,

S_n = a((1 - rⁿ)/(1 - r)) r ≠ 1

This can also written as

S_n = a((rⁿ - 1)/(r - 1))

Note: (1) S_n = a + ar + ar²+….. +ar^n-1

(2) If r = 1, then Sn = a + a + a +……………+a – na

(3) If r = 1, then formulas for S_n fail

(4) If r < 1, then use S_n = a((1 - rⁿ)/(1 - r))

(5) If r > 1, use S_n = a((rⁿ - 1)/(r - 1))

(6) Sum of an infinite G.P.,a + ar + ar² + ..... is a/(1 - r), - 1 r < 1

(7) If a, b, c are in G.P.then ‘b’ is the G.M. (geometric mean) between a and c, then b² = ac ⇒ b = √ac

(8) If A and G are A.M. and G.M. of two given distinct positive real numbers, then relation between A and G is A > G.

Sum of G.P. up to n terms if 1^st term is a and

the common ratio is r.

Sn =a.(1-rⁿ)/(1-r)………….(1) while r < 1 .

Sn =a.(r^n-1)/(r - 1)………(2) while r > 1 .

Sum up to infinite terms = a/(1-r).

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