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 displaystyle∑i=1n f(i)=363, then n is equal to .

Question

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 displaystyle∑i=1n f(i)=363, then n is equal to . (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

f(x+y)=f(x) f(y) put x=y=1 f(2)=(f(1))2=32 put x=2, y=1 f(3)=(f(1))3=33 vdots Similarly f(x)=3x displaystyle∑i=1n f(i)=363 ⇒ displaystyle∑i=1n 3i=363 (3+32+ ldots+3n)=363 (3(3n-1)/2)=363 3n-1=242 ⇒ 3n=243 ⇒ n=5

Q. 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 i=1∑n​f(i)=363, then n is equal to _____.

f(x+y)=f(x)f(y) put x=y=1f(2)=(f(1))2=32 put x=2,y=1f(3)=(f(1))3=33 ⋮ Similarly f(x)=3x i=1∑n​f(i)=363 ⇒i=1∑n​3i=363 (3+32+…+3n)=363 23(3n−1)​=363 3n−1=242⇒3n=243 ⇒n=5

Q. Suppose that a function $f : R \to R$ satisfies $f (x + y) = f (x) f (y)$ for all $x, y \in R$ and $f (1) = 3$ . If $i = 1 \sum n f (i) = 363,$ then $n$ is equal to _____.