retagged by
511 views

1 Answer

1 votes
1 votes
Given that, $ f(x+y) = f(x) + f(y) , f(1) = 2 $

And $, f(a+1) + f(a+2) + f(a+3) + \dots + f(a+n) = 16(2^{n}-1) \quad \longrightarrow (1) $

Now, put $n=1$, we get

$ f(a+1) = 16(2-1) = 16 $

$ \Rightarrow f(a) f(1) = 16 $

$ \Rightarrow f(a) \cdot 2 = 16 $

$ \Rightarrow f(a) = 8 \quad \longrightarrow (2) $

And, put $n=2$, we get

$ f(a+1) + f(a+2) = 16(2^{2} – 1) $

$ \Rightarrow f(a) f(1) + f(a) f(2) = 48 $

$ \Rightarrow 8 \cdot 2 + 8 \cdot f(2) = 48 $

$ \Rightarrow 16 + 8 f(2) = 48 $

$ \Rightarrow 8 f(2) = 32 $

$ \Rightarrow f(2) = 4 \quad \longrightarrow (3) $

And, put $n=3$, we get

$ f(a+1) + f(a+2) + f(a+3) = 16(2^{3}-1) $

$ \Rightarrow f(a) f(1) + f(a) f(2) + f(a) f(3) = 16(7) $

$ \Rightarrow 8 \cdot2 + 8 \cdot 4 + 8 f(3) = 112 $

$ \Rightarrow 16 + 32 + 8 f(3) = 112 $

$ \Rightarrow  48 + 8 f(3) = 112 $

$ \Rightarrow 8 f(3) = 64 $

$ \Rightarrow f(3) = 8 \quad \longrightarrow (4) $

On comparing equation $(2)$ and $(4)$, we get

$ \boxed{a=3} $

Correct Answer $: 3 $
edited by
Answer:

Related questions

1 votes
1 votes
1 answer
1
go_editor asked Mar 8, 2020
712 views
For any positive integer $n$, let $f(n)=n(n+1)$ if n is even, and $f(n)=n+3$ if n is odd. if $m$ is a positive integer such that $8f(m+1)-f(m)=2$, then $m$ equals ______...
1 votes
1 votes
1 answer
2
go_editor asked Mar 8, 2020
651 views
In a class, $60\%$ of the students are girls and the rest are boys. There are $30$ more girls than boys. If $68\%$ of the students, including $30$ boys, pass an examinati...
2 votes
2 votes
1 answer
3
go_editor asked Mar 8, 2020
578 views
Let $\text{S}$ be the set of all points $(x,y)$ in the $x-y$ plane such that $|x|+|y|\leq 2$ and $|x|\geq 1$. Then, the area, in square units, of the region represented b...
2 votes
2 votes
1 answer
4
go_editor asked Mar 8, 2020
593 views
With rectangular axes of coordinates, the number of paths from $(1,1)$ to $(8,10)$ via $(4,6)$, where each step from any point $(x,y)$ is either to $(x,y+1)$ or to $(x+1,...
2 votes
2 votes
1 answer
5
go_editor asked Mar 8, 2020
516 views
Let T be the triangle formed by the straight line $3x+5y-45=0$ and T the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as ...