Paragraph: Let f(x) be a function satisfying f(x+y)=f(x) f(y) for all x, y∈ ℕ such that f(1)=2 : Question: What is ∑_ x=1 ^6 2^x f(x) equal to?

Correct answer: D. 5460 aligned & ∑_ x=1 ^6 2^x f(x)=2 f(1)+2^2 f(2)+2^3 f(3) +2^6 f(6) & =2.(2)+4.(4)+8.(8)+ 64.64 & =4+16+64 6 terms & =(4(4^6-1))/(4-1)=(4 × 4095)/(3)=5460 aligned

NDA 2022

Exam: NDA
Year: 2022
Subject: Mathematics
Chapter: Sequences and Series

NDA 2022 · Mathematics · Sequences and Series

Paragraph: Let f(x) be a function satisfying f(x+y)=f(x) f(y) for...

Paragraph: Let

f(x)

be a function satisfying

f(x+y)=f(x) f(y)

for all

x, y\in \mathbb{N}

such that

f(1)=2

:Question: What is

\sum_{x=1}^6 2^x f(x)

equal to?

Options

A $1365$
B $2730$
C $4024$
D $5460$

Correct answer

D. $5460$

Step-by-step solution

\begin{aligned} & \sum_{x=1}^6 2^x f(x)=2 f(1)+2^2 f(2)+2^3 f(3) \ldots+2^6 f(6) \\ & =2.(2)+4.(4)+8.(8)+\ldots 64.64 \\ & =4+16+64 \ldots 6 \text{ terms } \\ & =\frac{4\left(4^6-1\right)}{4-1}=\frac{4 \times 4095}{3}=5460\end{aligned}

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