if f is a function satisfying …
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Naveen Sharma 6 years, 11 months ago
Ans. Given: f(x + y) = f(x)*f (y) for all x, y {tex}\in {/tex} N ......(1)
{tex}\displaystyle\sum_{i=1}^{n} f(x) = 120 \space \space ............(2){/tex}
f(1) = 3
Taking x = y = 1 in (1), we obtain
f(1 + 1) = f (2) = f (1) f (1) = 3 × 3 = 9
Similarly, f(3) = f (1 + 2) = f (1) f (2) = 3 × 9 = 27
f(4) = f(1 + 3) = f (1) f (3) = 3 × 27 = 81
f(1), f (2), f (3), …, that is 3, 9, 27, …, forms a G.P.
First term is 3 and common ratio is also 3.
Also we know
{tex}=> S_n = {a(r^n -1)\over r - 1}{/tex}
{tex}=> S_n = {3(3^n -1)\over 3 - 1}{/tex} .................... (3)
From (2) and (3), We get
{tex}=> 120 = {3(3^n -1)\over 3 - 1}{/tex}
{tex}=> 120 = {3(3^n -1)\over 2}{/tex}
{tex}=> 80 = 3^n -1{/tex}
{tex}=> 81 = 3^n {/tex}
{tex}=> 3^4 = 3^n {/tex}
n = 4,
So Value of n = 4
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