Test Index

UPSC NDA 2 2022 Mathematics Solved Paper

Question Numbers: 31-33

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 : 32 of 120

Marks: +1, -0

What is

\displaystyle\sum\limits_{x=1}^{5}

f(2x − 1) equal to ?

Solution:

Concept:

The functional equation

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

for all

x,y\in\mathbb{N}

implies an exponential function.

Explanation:

Given

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

and

f(1)=2

, the function must be of the form

f(x)=a^x

Using

f(1)=2

, we get

a^1=2

, so

a=2

. Hence

f(x)=2^x

We need

\displaystyle\sum_{x=1}^{5} f(2x-1) = \sum_{x=1}^{5} 2^{2x-1}

Compute each term:

For

x=1

2^{2\cdot1-1}=2^{1}=2

x=2

2^{3}=8

x=3

2^{5}=32

x=4

2^{7}=128

x=5

2^{9}=512

Sum:

2+8+32+128+512 = 682

Answer:

682

(option B).

Go to Question:

Prev Question

Next Question