CAT Quantitative Aptitude Questions | CAT Algebra - Progressions questions

The question is about Sum of a progression. A term is given and we need to find out the order of the term in the series. 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 12: If S_n = n³ + n² + n + 1 , where S_n denotes the sum of the first n terms of a series and t_m = 291, then m is equal to?

1. 24
2. 30
3. 26
4. 10

Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Progressions: How you would form the equation for this one?

S_n - S_n = t_n
Substitute m instead of n
S_m - S_m-1 = t_m
We know that S_n = n³ + n² + n + 1

Hence m³ + m² + m + 1
m³ + m² + m + 1 - [(m-1)³ + (m-1)² + (m-1) + 1 ] = 291
m³ + m² + m + 1 - [m³ - 3m² + 3m - 1 + m² - 2m + 1 + m - 1 + 1] = 291
1 + 3m² + 3m + 1 -2m - 1 = 291
-3m² + m - 290 = 0
3m² - m + 290 = 0

Solving above equation we get m = 10, \\frac{-29}{3}\\)
M cannot be negative as well as decimal as m is the term of the sequence.
Hence m = 10

The question is "If S_n = n³ + n² + n + 1 , where S_n denotes the sum of the first n terms of a series and t_m = 291, then m is equal to?"

Hence the answer is "10"

Choice D is the correct answer.