Suppose that a function f : R → R satisfies
f(x + y) = f(x)f(y) for all x, y ∈ R and f(1) = 3.
If Σᵢ₌₁ⁿ f(i) = 363 then n is equal to ________ .

Solution

Maths sir · 4 steps

Turn the product rule into powers of 3, and the sum becomes a plain geometric series.

1. Find f at integers. Put x = y = 1: f(2) = f(1)f(1) = 3². Repeating this gives f(n) = f(n-1)f(1) = 3ⁿ for every positive integer n.
2. Write the sum. The sum is 3 + 3² + … + 3ⁿ. That is a GP with first term 3 and ratio 3.

   Σᵢ₌₁ⁿ 3ⁱ = 3(3ⁿ - 1)/(3 - 1) = 363

3. Solve for n. Multiply both sides by 2/3: 3ⁿ - 1 = 242, so 3ⁿ = 243 = 3⁵.
4. So. n = 5. Check: 3 + 9 + 27 + 81 + 243 = 363. The answer is 5.

The idea to keep. Whenever f(x+y) = f(x)f(y) and f(1) = a, you get f(n) = aⁿ at integers. Any sum of f(i) is then just a geometric series.

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