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 $n=$ ?}

• A. 8
• B. 9
• C. 10
• D. 11
• E. None of the above

Hint:

Whenever $(x+y)=(x)(y)$ is given, the function usually grows exponentially. Check for powers of 2 or 3 by computing small values and then generalize.

Answer 1

The Correct Option is B

Step 1: Define the function.
Let $f(x)=(x)$. Given: \[ f(1)=2, \quad f(x+y)=f(x)\cdot f(y) \] Step 2: Compute initial terms.
- For $x=2$: $f(2)=f(1+1)=f(1)\cdot f(1)=2\cdot2=4$.
- For $x=3$: $f(3)=f(2+1)=f(2)\cdot f(1)=4\cdot2=8$.
- For $x=4$: $f(4)=f(3+1)=f(3)\cdot f(1)=8\cdot2=16$. So the pattern emerges: \[ f(n)=2^n \] Step 3: Express the summation.
\[ \sum_{x=1}^{n} f(x)=2^1+2^2+2^3+\ldots+2^n \] This is a geometric progression with first term $a=2$, ratio $r=2$, $n$ terms. Step 4: Sum of G.P.
\[ S_n=\frac{a(r^n-1)}{r-1}=\frac{2(2^n-1)}{1}=2^{n+1}-2 \] We are given $S_n=1022$. \[ 2^{n+1}-2=1022 \quad \Rightarrow \quad 2^{n+1}=1024 \] \[ 2^{n+1}=2^{10} \quad \Rightarrow \quad n+1=10 \quad \Rightarrow \quad n=9 \] Final Answer: \[ \boxed{9} \]