Consider a function f satisfying f(x + y) = f(x) f(y) where x, y are positive integers and f(1) = 2. If f(a + 1) + f(a + 2) +…+ f(a + n) = 16(2 n – 1) then a is equal to

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Consider a function f satisfying f(x + y) = f(x) f(y) where x, y are positive integers and f(1) = 2. If f(a + 1) + f(a + 2) +&hellip;+ f(a + n) = 16(2 n &ndash; 1) then a is equal to

Accepted Answer

3 — Easy The multiplicative relation $f(x+y)=f(x)f(y)$ with $f(1)=2$ forces $f(n)=2^{n}$. The given sum is then a geometric series; factor out $2^{a+1}$ and match it to $16=2^{4}$. 1 Identify $f$. Build up from $f(1)=2$. $$\begin{aligned} &f(2)=f(1)f(1)=2\cdot 2=2^2\ &f(3)=f(1)f(2)=2\cdot 2^2=2^3\ &\Rightarrow\ f(n)=2^{\,n} \end{aligned}$$ 2 Sum as a geometric series. Write each term with $f(n)=2^n$ and factor $2^{a+1}$. $$\begin{aligned} &\sum_{k=1}^{n} f(a+k)=2^{a+1}+2^{a+2}+\dots+2^{a+n}\quad

CAT 2019 Slot 1 — QA Question 33

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