JEE PYQ: Sequence And Series Question 26
Question 26 - 2020 (04 Sep Shift 1)
If $1 + (1 - 2^2 \cdot 1) + (1 - 4^2 \cdot 3) + (1 - 6^2 \cdot 5) + \ldots + (1 - 20^2 \cdot 19) = \alpha - 220\beta$, then an ordered pair $(\alpha, \beta)$ is equal to:
(1) $(10, 97)$
(2) $(11, 103)$
(3) $(10, 103)$
(4) $(11, 97)$
Show Answer
Answer: (2)
Solution
The given series is $1 + (1 - 2^2 \cdot 1) + (1 - 4^2 \cdot 3) + (1 - 6^2 \cdot 5) + \ldots + (1 - 20^2 \cdot 19)$
$S = 1 + \sum_{r=1}^{10}[1 - (2r)^2(2r-1)]$
$= 1 + 10 - \sum_{r=1}^{10}(8r^3 - 4r^2) = 1 + 10 - 8\left(\frac{10 \times 11}{2}\right)^2 + 4 \times \frac{10 \times 11 \times 21}{6}$
$= 11 - 2 \times (110)^2 + 4 \times 55 \times 7$
$= 11 - 220(110-7)$
$= 11 - 220 \times 103 = \alpha - 220\beta$
$\Rightarrow \alpha = 11, \beta = 103$
BETA
