Test Index

Functions Part 2

Section: Mathematics

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Question : 14 of 73

Marks: +1, -0

Let

f

g

be functions satisfying

f(x+y)=f(x) f(y), f(1)=7

g(x+y)=g(xy), g(1)=1

\sum\limits_{x=1}^{n} \left( \frac{f(x)}{g(x)} \right) =19607

\sum\limits_{x=1}^{ n } (\;{f(x)}/{ g (x)}) =19607

, then n is equal to

[22 Jan 2026 Shift 2]

6
7
4
5

Solution:

Concept:

Functional equations define exponential and constant functions.

Explanation:

From

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

and

f(1)=7

, we deduce

f(x)=7^x

.

From

g(x+y)=g(xy)

put

y=1

:

g(x+1)=g(x)

, so

g

is constant. Since

g(1)=1

, we get

g(x)=1

for all

x

.

The given sum is

\sum_{x=1}^n \frac{f(x)}{g(x)} = \sum_{x=1}^n 7^x = 19607

.

Sum of geometric series:

7\frac{7^n-1}{7-1}=19607

.

Multiply both sides by

6

:

7(7^n-1)=117642

→

7^n-1=16806

→

7^n=16807

.

Since

7^5=16807

, we get

n=5

.

Answer:

n=5

, which corresponds to option D.

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