The question is about sum of N terms of an Arithmetic Progression. If modulus of two terms in an AP is equal, how we can determine whether the series is increasing or decreasing? For that we need more details. 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. CAT Exam tests the idea of progression often in the CAT Quantitative Aptitude section and this could also be tested in DI LR section of the CAT Exam as a part of a puzzle.

Question 4: Let the n^{th} term of AP be defined as t_{n}, and sum up to 'n' terms be defined as S_{n}. If |t_{8}| = |t_{16}| and t_{3} is not equal to t_{7}, what is S_{23}?

- 23(t
_{16}- t_{8}) - 0
- 23t
_{11} - Cannot be determined

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|t_{8}|=|t_{16}|. This can happen under two scenarios t_{8} = t_{16} or t_{8} = – t_{16}.

If t_{8} = t_{16}, the common difference would be 0 suggesting that t_{3} would be equal to t_{7}.

However, we know t_{3} is not equal to t_{7}, so the common difference cannot be zero.

This tells us that t_{8} = – t_{16} Or, t_{8} + t_{16} = 0.

If t_{8} + t_{16} = 0, then t_{12} = 0. t_{12} = t_{8} + 4d, and t_{16} – 4d So, t_{12} = \\frac{t_8+t_16}{2}\\)

For any two terms in an AP, the mean is the term right in between them.

So, t_{16} is the arithmetic mean of t_{8} and t_{16}.

So, t_{12} = 0.

Now, S_{23} = 23 * t_{12}. We know that average of n terms in an A.P. is the middle term.

This implies that sum of n terms in an A.P., is n times the middle term. So, S_{23} = 0.

The question is **"If |t _{8}| = |t_{16}| and t_{3} is not equal to t_{7}, what is S_{23}?"**

Choice B is the correct answer.

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