CAT Quantitative Aptitude Questions | CAT Algebra - Progressions questions

CAT Questions | Algebra | Arithmetic Progressions - Sum of N terms

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 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

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Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Progressions: What is the value of the middle term of two terms which gives zero when we add them?

|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, t16 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.

The question is "If |t8| = |t16| and t3 is not equal to t7, what is S23?"

Hence the answer is "zero."

Choice B is the correct answer.

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