Let , where the function satisfies for all natural numbers , and . Then, the natural number '' is
- A.2
- B.4
- C.3
- D.16
Show answer and solution
Correct option: C
Solution
f(x+y) = f(x)f(y) with f(1) = 2 means f(n) = 2n for all natural numbers n.
So f(a+k) = 2^(a+k), and the sum becomes:
∑(k=1 to 10) 2^(a+k) = 2a · (2¹ + 2² + ... + 2¹⁰) = 2a · 2(2¹⁰ - 1)/(2-1) = 2^(a+1) · (2¹⁰ - 1)
Setting equal to 16(2¹⁰ - 1):
2^(a+1) = 16 = 2⁴ → a+1 = 4 → a = 3.
So f(a+k) = 2^(a+k), and the sum becomes:
∑(k=1 to 10) 2^(a+k) = 2a · (2¹ + 2² + ... + 2¹⁰) = 2a · 2(2¹⁰ - 1)/(2-1) = 2^(a+1) · (2¹⁰ - 1)
Setting equal to 16(2¹⁰ - 1):
2^(a+1) = 16 = 2⁴ → a+1 = 4 → a = 3.
[JEE Main 2019 · 9 Apr · Shift II]