# Arithmetic and Geometric Progressions

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## AP: Common Difference

Let the nth term of AP be defined as tn, and sum up to 'n' terms be defined as Sn. If |t8| = |t16| and t3 is not equal to t7, what is S23?
1. 23(t16 - t8)
2. 0
3. 23t11
4. Cannot be determined

Choice B. 0

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## Detailed Solution

|t8|=|t16|. This can happen under two scenarios t8 = t16 or t8 = – t16.

If t8 = t16, the common difference would be 0 suggesting that t3 would be equal to t7. However, we know t3 is not equal to t7, so the common difference cannot be zero.

This tells us that t8 = – t16 Or, t8 + t16 = 0.

If t8 + t16 = 0, then t12 = 0. t12 = t8 + 4d, and t16 – 4d So, t12 =

• ${\frac{ t_{8}+t_{16} }{2}}$

• . For any two terms in an AP, the mean is the term right in between them. So, t12 is the arithmetic mean of t8 and t16.

So, t12 = 0.
Now, S23 = 23 × t12. 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, S23 = 0. Correct Answer: zero.

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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.