CAT 2021 Question Paper | Quant Slot 2

CAT 2021 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 20 : For a sequence of real numbers x₁, x₂, ..., x_n, if x₁ - x₂ + x₃ - ... + (-1)^{n + 1}x_n = n² + 2n for all natural numbers n, then the sum x₄₉ + x₅₀ equals

Video Explanation

Explanatory Answer

We are given a generic formula in ‘n’ for the sum of first n terms, with alternating signs.
S_n = x₁ - x₂ + x₃ - x₄ + ….. + (-1)ⁿ⁺¹ x_n = n² + 2n = n(n + 2)
(Observe that the even terms have a negative sign, while the odd terms have a positive sign)
S₅₀ = 50 (50 + 2) = 50 (52) = 50 (2) (26) = 2600
S₄₉ = 49 (49 + 2) = 49 (51) = 2499
S₄₈ = 48 (48 + 2) = 48 (50) = (24) (2) (50) = 2400
S₅₀ = S₄₉ - x₅₀
x₅₀ = S₄₉ - S₅₀ = 2499 - 2600 = -101
S₄₉ = S₄₈ + x₄₉
x₄₉ = S₄₉ - S₄₈ = 2499 - 2400 = 99
x₄₉ + x₅₀ = 99 - 101 = -2
Alternate Solution:
Observe that n² + 2n is an increasing function for all natural numbers.
That is, if x > y; x² + 2x > y² + 2y.
S_n = n² + 2n = n(n + 2) = n² + 2n
S_{n - 1} = (n - 1)(n +1) = n² - 1
S_n - S_n-1 = n² + 2n - (n² - 1) = 2n + 1
So adding the n^th term to the sequence increases the sequence by 2n + 1
The even terms of the sequence are subtracted, but then the sum of the terms in the sequence still increases, this means that the n^th term is negative if n is even and n^th term is positive if n is odd, but the magnitude of the n^th term is is always (2n +1)
So, x_n = (-1)ⁿ⁺¹ (2n + 1)
x₅₀ = (-1)⁵⁰⁺¹ (2(50) + 1) = -101
x₄₉ = (-1)⁴⁹⁺¹ (2(49) + 1) = 99
x₄₉ + x₅₀ = 99 - 101 = -2

The answer is '-2'

Choice B is the correct answer.