Misc 7 - If f (x + y) = f(x) f(y) such that  f(1) = 3, find n - Miscellaneous

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  1. Chapter 9 Class 11 Sequences and Series
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Misc 7 If f is a function satisfying f (x + y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and , find the value of n. Given that : f (x + y) = f(x) f(y) ∀ x, y ∈ N and f(1) = 3 ∴ f (1) = 3 f (2) = 9 = 3﷮2﷯ f (3) = 27 = 3﷮3﷯ f (4) = 81 = 3﷮4﷯ Similarly, f (5) = 3﷮5﷯ f (6) = 3﷮6﷯ Thus our series is 3, 3﷮2﷯, 3﷮3﷯, 3﷮4﷯, ………… n terms This is a GP, where a = 3 r = 3﷮2﷯﷮3﷯ =3 Given sum of GP = 120 = 𝑎( 𝑟﷮𝑛﷯ − 1)﷮𝑟 −1﷯ Putting a = 3, r = 3 & sum = 120 120 = 3( 3﷮𝑛﷯ −1)﷮3 − 1﷯ ⇒ 3( 3﷮𝑛﷯ −1)﷮2﷯ = 120 ⇒ 3﷮𝑛﷯ −1 = 120 × 2﷮3﷯ ⇒ 3﷮𝑛﷯ −1 = 40 × 2 3﷮𝑛﷯ = 80 3﷮𝑛﷯ = 80 + 1 3﷮𝑛﷯ = 80 3﷮𝑛﷯ = 3﷮4﷯ n = 4 ∴ n = 4

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