# Arithmetic and Geometric Progressions

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It appears that we are adding two additional terms, but the sum does not appear to change. When would this be the case?

## GP: Common Ratio

Sum of first 12 terms of a GP is equal to the sum of the first 14 terms in the same GP. Sum of the first 17 terms is 92, what is the third term in the GP?
1. 92
2. -92
3. 46
4. 231

Choice A. 92

## Detailed Solution

Sum of first 12 terms is equal to sum of first 14 terms.
Sum of first 14 terms = Sum of first 12 terms + 13th term + 14th term
=> 13th term + 14th term = 0

Let us assume 13th term = k, common ratio = r. 14th term will be kr.
k + kr = 0
k (1 + r) = 0
=> r = -1 as k cannot be zero

Common ratio = -1.
Now, if the first term of this GP is a, second term would be -a, third would be a and so on
The GP would be a, -a, a, -a, a, -a,...

Sum to even number of terms = 0
Sum to odd number of terms = a

Sum to 17 terms is 92 => a = 92
Third term = a = 92

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## More questions from Progressions

With some simple but very powerful ideas, one can cut down on a lot of working when it comes to progressions. For example, anchoring a progression around its middle term can be very useful. Reinforce these ideas with these questions.