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JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Sequences and Series

Let

f (x)

be a function such that

f (x + y) = f (x) \cdot f (y)

for all

x, y \in N

. If

f (1) = 3

and

\sum_{k = 1}^{n} f (k) = 3279

, then the value of n is

(A)6
(B)8
(C)7✓ Correct
(D)9

Answer: (C)

Solution

The relation

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

with

f (1) = 3

forces

f (k) = 3^{k}

for every natural

k

. The sum is geometric:

\sum_{k = 1}^{n} 3^{k} = \frac{3 ( 3 ^{n} - 1 )}{2} = 3279

, so

3^{n} - 1 = 2186

and

3^{n} = 2187 = 3^{7}

. Thus

n = 7

.

JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Single correct option

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