Suppose f is a function satisfying f(x+y) = f(x) + f(y) for all x, y ∈ N and f(1) = 1/5. If sum_n=1^m (f(n))/(n(n+1)(n+2)) = 1/12, then m is equal to _______

Suppose f is a function satisfying f(x+y) = f(x) + f(y) for… — JEE Main 2023 Mathematics PYQ

JEE Main 2023 · 29 Jan · Shift 1 · numerical answer

Suppose

f

is a function satisfying

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

for all

x, y \in \mathbb{N}

and

f(1) = \frac{1}{5}

. If

\sum_{n=1}^{m} \frac{f(n)}{n(n+1)(n+2)} = \frac{1}{12}

, then

m

is equal to _______

Show answer and solution

Answer: 10

Solution

From the functional equation, f(n) = n·f(1) = n/5.

Substituting: sum = Σ(n=1 to m) of (n/5)/(n(n+1)(n+2)) = (1/5)·Σ 1/((n+1)(n+2)).

Using partial fractions: 1/((n+1)(n+2)) = 1/(n+1) - 1/(n+2), so the sum telescopes to 1/2 - 1/(m+2).

Setting equal to 1/12: (1/5)·(1/2 - 1/(m+2)) = 1/12 → 1/2 - 1/(m+2) = 5/12 → 1/(m+2) = 1/12 → m+2 = 12 → m = 10.

[JEE Main 2023 · 29 Jan · Shift 1]

Practise Sequences and Series the way it is asked

A timed set of real JEE Main Sequences and Series questions, graded as the exam grades. No sign-up.

Start a Sequences and Series test

More: all JEE Main Sequences and Series questions · JEE Main 2023 · Mathematics chapter list

JEE Main 2023 Mathematics question on Sequences and Series