# CAT 2021 Question Paper | Quant Slot 2

###### CAT Previous Year Paper | CAT Quant Questions | Question 20

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, ..., xn, if x1 - x2 + x3 - ... + (-1)n + 1xn = n2 + 2n for all natural numbers n, then the sum x49 + x50 equals

1. 2
2. -2
3. 200
4. -200

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We are given a generic formula in ‘n’ for the sum of first n terms, with alternating signs.
Sn = x1 - x2 + x3 - x4 + ….. + (-1)n+1 xn = 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.
Sn = n2 + 2n = n(n + 2) = n2 + 2n
Sn - 1 = (n - 1)(n +1) = n2 - 1
Sn - Sn-1 = n2 + 2n - (n2 - 1) = 2n + 1
So adding the nth 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 nth term is negative if n is even and nth term is positive if n is odd, but the magnitude of the nth term is is always (2n +1)
So, xn = (-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

Choice B is the correct answer.

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