Is he wrong ? I I II

Number Theory Level 3

A student wrote an equation on a blackboard. He then changed a number with a letter. If he wrote ( 2 + 1 ) + ( 2 + 2 ) 2 + ⋯ + ( 2 + a ) a = 2 a + 4 a (2+1)+(2+2)^2+\dots+(2+a)^a=2^a+4^a such that a a ∈ N \in \mathbb{N} .

If ( 2 + a ) (2+a) is of odd parity then, is he wrong ?

Yes No

Mohd. Hamza by Mohd. Hamza , 18, India

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1 solution

Mr. India
Mar 24, 2019

2 + a = o d d ⇒ a = o d d 2+a=odd\Rightarrow a=odd

Last term of equation is ( 2 + a ) a (2+a)^a

As a a is o d d odd → 2 + a = o d d → ( 2 + a ) a = o d d \rightarrow 2+a=odd\rightarrow (2+a)^a=odd

So, LHS is odd but RHS is even.

So, he definitely has made mistake

2 Helpful 0 Interesting 1 Brilliant 0 Confused

I have made little changes in my problem. So please change your solution accordingly.

Mohd. Hamza - 2 years, 2 months ago

As the problem is currently written, the LHS could be odd or even.

Jordan Cahn - 2 years, 2 months ago

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It will be always even now. Will edit in some time

Mr. India - 2 years, 2 months ago

Am I misunderstanding the question?

Let a = 1 a = 1 . a ∈ N a\in\mathbb N and 2 + a 2+a is odd.

But 2 + 1 ≠ 2 1 + 4 1 2+1\neq 2^1 + 4^1

Micah Wood - 2 years, 2 months ago

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