If \sum_{x=2}^{n}f(x)=2044, then what is the value of n?
Consider the following for the next three (03) items that follow : Let f(x) be a function satisfying f(x+y)=f(x)f(y) for <strong>ALL</strong> x, y\in N such that f(1)=2 :
- A. 8
- B. 9
- C. 10 ✓
- D. 11
Correct Answer: C. 10
Explanation
The functional equation f(x+y)=f(x)f(y) with f(1)=2 implies f(x) = 2^x. The sum is a geometric progression: \sum_{x=2}^{n}2^x = 2^2 + 2^3 + \dots + 2^n. The sum of this GP is 4\frac{2^{n-1}-1}{2-1} = 2044. Simplifying gives 2^{n-1}-1 = 511, so 2^{n-1} = 512 = 2^9. Therefore, n-1 = 9 \implies n = 10.
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