Question

Given a G.P. with a = 729 and 7^th term 64, determine S₇.

Answer 1

Given:

First term, a = 729
7^th term = 64

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Step 1: Find the common ratio (r)

The n^th term of a G.P. is given by:

T_n = a rⁿ⁻¹

For the 7^th term:

64 = 729 · r⁶

r⁶ = 64 / 729

r⁶ = (2⁶) / (3⁶)

∴ r = 2 / 3

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Step 2: Use the formula for the sum of first 7 terms

S_n = a (1 − rⁿ) / (1 − r),   r ≠ 1

So,

S₇ = 729 (1 − (2/3)⁷) / (1 − 2/3)

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Step 3: Simplify

1 − 2/3 = 1/3

∴

S₇ = 729 × 3 × [1 − (2/3)⁷]

S₇ = 2187 [1 − 128/2187]

S₇ = 2187 − 128

S₇ = 2059

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Final Answer:

The sum of the first 7 terms of the G.P. is
S₇ = 2059