Given, f(x+y)=2f(x)f(y) and f(1)=2 For x=1 and y=1, f(1+1)=2f(1)f(1) ⇒f(2)=2(f(1))2=2(2)2=23 For x=1,y=2 f(1+2)=2f(1)y(2) ⇒f(3)=2⋅2⋅23=25 For x=1,y=3 f(1+3)=2f(1)f(3) ⇒f(4)=2⋅2⋅25=27 For x=1,y=4 f(1+4)=2f(1)f(4) ⇒f(5)=2.2⋅27=29.... Also given
10
∑
k=1
f(α+k)=
512
3
(220−1) ⇒f(α+1)+f(α+2)+f(α+3)+...+f(α+10)=
512
3
(220−1) ⇒f(α+1)+f(α+2)+f(α+3)+...+f(α+10)=
29((22)10−1)
22−1
This represent a G.P with first term =29 and common ratio =22 ∴ First term =f(α+1)=29....(2) From equation (1),f(5)=29 ∴ From (1) and (2), we get
f(α+1)=29=f(5) ⇒f(α+1)=f(5) ⇒f(α+1)=f(4+1) Comparing both sides we get, α=4