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if f is a function satisfying …

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if f is a function satisfying f(x+y)=f(x) f(y) for all x,y belongs to n such that f(1)=3 and sigma x =1 to n f(x)=120 then find n.

  • 1 answers

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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