Answered: Theorem: A positive integer n will be…

Math

Advanced Math

Theorem: A positive integer n will be the sum of a sequence of consecutive positive integers if and only if n #2^n.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

I'm not sure where to go from here. We did a problem on number from 1-35 and finding all the ways to find all the possible positive consecutive integer numbers for each and then we have to write a proof on this. And I know that if it's and if and only if statement it needs to be proved both ways but I'm Not sure how to do it. I started it this way

Theorem: A positive integer n will be the sum of a
sequence of consecutive positive integers if and only if n
#2^n.
Proof:
Let s be the smallest of c consecutive integers € Z + such
that c≥2.
With respect to the Sum of Arithmetic Sequence, a + c =
c( 2s+c-1)2.
Proposed contradiction:
Let c( 2s +c-1)2 be a power of 2.
Then c and 2s + c - 1 are powers of 2.
Since c≥ 2, c is even.
Hence, 2s + c - 1is odd.
Since 2s + c - 1 is odd and a power of 2, 2s +c-1=1
However, 2s + c +1 > 2(0) +2 -1 = 1.
This is a contradiction.
Then, any sum of two or more consecutive integers
cannot be 2n.

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