for n=2,3,4....

prove that given x1x2...xn=1,

then x1+x2+x3+...+xn=n iff x1=x2=x3=...=xn=1

I tried to prove by induction. While the "if" direction is obvious, but I am kind of stuck in the "only if" direction.

Here's my take.

for n=2 given x1x2=1

only if: x1+x2=2, we have x1=2-x2 (2-x2)x2=1 x2=1 then x1 also =1

if: x1=x2=1 then, clearly x1+x2=2

for n=k assume x1x2...xk=1 then

x1+x2+x3+...+xk=k iff x1=x2=x3=...=xk=1

for n=k+1

I am stuck here! please help



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