If f(x + y) = f(x)f(y) and Σₓ₌₁^∞ f( x ) = 2 , x, y ∈ N, where N is the set of all natural number, then the value of f( 4 )/f( 2 ) is :

Solution

Maths sir · 4 steps

f(x + y) = f(x)f(y) on naturals means f(x) is a power of f(1), so the sum is a geometric series.

1. Find the form. Let f(1) = a. Then f(2) = f(1)f(1) = a², f(3) = f(2)f(1) = a³, and in general f(x) = aˣ for every natural x.
2. Use the sum. The sum is the geometric series a + a² + a³ + …. It converges only for |a| < 1, and then it equals a/(1-a).

   a/(1-a) = 2 ⇒ a = 2 - 2a ⇒ a = 2/3

3. Take the ratio. f(4)/f(2) = a⁴/a² = a² = (2/3)² = 4/9.
4. So. The value is 4/9, option (d).

The idea to keep. Whenever f(x + y) = f(x)f(y) on natural numbers, write f(x) = aˣ with a = f(1). Every sum or ratio then becomes a power or a geometric series.

Answer from the NTA final answer key. Solution by PrepWiser.