CAT Quantitative Aptitude Questions | CAT Algebra - Progressions questions

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 |t₈| = |t₁₆| and t₃ is not equal to t₇, what is S₂₃?

1. 23(t₁₆ - t₈)
2. 0
3. 23t₁₁
4. Cannot be determined

Video Explanation

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.