InterviewSolution
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InterviewSolution
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If f is a function satisfying `f(x+y)=f(x)f(y)`for all `x ,y in X`such that `f(1)=3`and `sum_(x=1)^nf(x)=120`, find the value of n. |
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Answer» We know, `a^(m+n) = a^ma^n` It means, `f(x)` can be `a^x` to satisfy the given condition. Then, `f(x+y) = f(x)f(y)`. `:. f(x) = a^x` We are given, `f(1) = 3 => a^1 = 3=> a = 3` `f(2) = 3^2 = 9` `f(3) = 3^3 = 27` So, the series will be, `3,9,27...` It will form a GP with `a = 3 and r = 3` Now, Sum of first n terms in a GP, `S_n = (a(r^n -1)/(r-1))` We are given, `S_n = 120` `:. (3(3^n -1))/(3-1) = 120=>3^n-1 = 80=> 3^n = 81` So, value of `n` is `4`.`=>3^n = 3^4=> n = 4` |
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