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# Is the 22nd term of an Arithmetic Progression odd?

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Intern
Joined: 20 Oct 2018
Posts: 39
Is the 22nd term of an Arithmetic Progression odd?  [#permalink]

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04 Aug 2019, 06:28
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Difficulty:

75% (hard)

Question Stats:

36% (01:31) correct 64% (00:55) wrong based on 39 sessions

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Is the 22nd term of an Arithmetic Progression odd?

1. The 24th tern is even
2. The first term is odd

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GMAT Tutor
Joined: 24 Jun 2008
Posts: 2303
Re: Is the 22nd term of an Arithmetic Progression odd?  [#permalink]

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05 Aug 2019, 04:53
1
Is the 22nd term of an Arithmetic Progression odd?
1. The 24th tern is even
2. The first term is odd

The OA is given as A here, which is not correct. Using only Statement 1, the 22nd, 23rd and 24th terms of the sequence could be, say

22, 23, 24

in which case the 22nd term is even, but they could also be

21, 22.5, 24

in which case the 22nd term is odd.

As the question is written, the answer is C, since with both statements you can be certain the 22nd term is not odd. If you pretend the 22nd term is odd, you can see you'll reach a contradiction. If the list goes up by d from one term to the next:

- if the 22nd and 24th terms are integers in an equally spaced list, then 2d must be an integer, and every term in an even position in the sequence must be an integer
- so the 2nd term would then be an integer
- but we know the first term is an integer too, and if the 1st and 2nd terms are integers, every term is an integer, and the spacing d is an integer
- but then the 22nd and 24th terms would need to both be even or both be odd, since they are 2d apart. But they're not both even or both odd, so this entire situation is impossible.

So it's impossible, using both Statements, for the 22nd term to be odd, and the answer is C. The reasoning here is probably a lot more complicated than the question designer intended, if the question designer believed the answer was A. The answer is only A if the question tells you the terms in the sequence are integers.
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Re: Is the 22nd term of an Arithmetic Progression odd?  [#permalink]

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04 Aug 2019, 16:02
Assume the first term =A
Common difference= k

Statement 1-
The 24th term of an Arithmetic Progression= A+23k=Even

The 22nd term of an Arithmetic Progression= A+21k=(A+23k)-2k
A+23k is even, and 2k is always even

Hence, The 22nd term of an Arithmetic Progression is also even.

$$Sufficient$$

Statement 2-
A is odd
But we have no idea about whether k is even of odd.

Insufficient

Is the 22nd term of an Arithmetic Progression odd?

1. The 24th tern is even
2. The first term is odd

Posted from my mobile device
Intern
Joined: 10 Aug 2017
Posts: 5
Re: Is the 22nd term of an Arithmetic Progression odd?  [#permalink]

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05 Aug 2019, 05:09
This question is absolutely not GMAT-like. GMAC won't ask with such specific term "arithmetic progression".

Don't bother to answer the question

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Joined: 19 Oct 2018
Posts: 1991
Location: India
Re: Is the 22nd term of an Arithmetic Progression odd?  [#permalink]

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05 Aug 2019, 07:56
IanStewart I missed the subtle case. Thanks for the explanation.

IanStewart wrote:
Is the 22nd term of an Arithmetic Progression odd?
1. The 24th tern is even
2. The first term is odd

The OA is given as A here, which is not correct. Using only Statement 1, the 22nd, 23rd and 24th terms of the sequence could be, say

22, 23, 24

in which case the 22nd term is even, but they could also be

21, 22.5, 24

in which case the 22nd term is odd.

As the question is written, the answer is C, since with both statements you can be certain the 22nd term is not odd. If you pretend the 22nd term is odd, you can see you'll reach a contradiction. If the list goes up by d from one term to the next:

- if the 22nd and 24th terms are integers in an equally spaced list, then 2d must be an integer, and every term in an even position in the sequence must be an integer
- so the 2nd term would then be an integer
- but we know the first term is an integer too, and if the 1st and 2nd terms are integers, every term is an integer, and the spacing d is an integer
- but then the 22nd and 24th terms would need to both be even or both be odd, since they are 2d apart. But they're not both even or both odd, so this entire situation is impossible.

So it's impossible, using both Statements, for the 22nd term to be odd, and the answer is C. The reasoning here is probably a lot more complicated than the question designer intended, if the question designer believed the answer was A. The answer is only A if the question tells you the terms in the sequence are integers.
Re: Is the 22nd term of an Arithmetic Progression odd?   [#permalink] 05 Aug 2019, 07:56