Question

The operation $(x)$ is defined by (i) $(1) = 2$ (ii) $(x + y) = (x) \cdot (y)$ for all positive integers x and y. If $\sum_{x=1}^{n} (x) = 1022$ then $n =$

Accepted Answer

B. From the functional equation, $f(x) = 2^x$. $$\sum_{x=1}^{n} 2^x = 2 + 2^2 + 2^3 + ... + 2^n = \frac{2(2^n - 1)}{2 - 1} = 2^{n+1} - 2$$ Given $2^{n+1} - 2 = 1022 \implies 2^{n+1} = 1024 = 2^{10} \implies n + 1 = 10 \implies n = 9$.

Function Operation Summation

Explanation

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