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JEE Main · 2020 · 06 Sep 2020 · Shift 2 · Sequences and Series

Suppose that a function

f : R \to R

satisfies

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

for all

x, y \in R

and

f (1) = 3

. If

\sum_{i = 1}^{n} f (i) = 363

, then

n

is equal to

Answer: 5

Solution

The relation

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

with

f (1) = 3

forces

f (x) = 3^{x}

. The required sum is the geometric series

\sum_{i = 1}^{n} 3^{i} = \frac{3 ( 3 ^{n} - 1 )}{3 - 1} = 363

, so

3^{n} - 1 = 242

, giving

3^{n} = 243 = 3^{5}

and

n = 5

.

JEE Main · 2020 · 06 Sep 2020 · Shift 2 · Integer answer

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