Proving with a sequence The question is : Show that if n is a power of 2, then sum_(i=0)^(log_2n-1)2^i=n-1.Tried induction at first and tried to prove it on 2n but nothing came out of it. Then i tried every possible way with a series and i'm close but still can't prove it.

shadyufog0

shadyufog0

Answered question

2022-09-01

Proving with a sequence
The question is :
Show that if n is a power of 2, then
i = 0 log 2 n 1 2 i = n 1 .
Tried induction at first and tried to prove it on 2n but nothing came out of it. Then i tried every possible way with a series and i'm close but still can't prove it.
Thanks in advance !

Answer & Explanation

bewagox7

bewagox7

Beginner2022-09-02Added 10 answers

If n is a power of 2 then you have that n = 2 m for some m N . Thus
log 2 n = log 2 2 m = m log 2 2 = m
and therefore
i = 0 ( log 2 n ) 1 2 i = i = 0 m 1 2 i = Geometric sum = 1 2 m 1 + 1 1 2 = 1 2 m 1 = 2 m 1
but since 2 m = n, the last term is equal to n 1

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