Question

If $1+\left(1-2^{2} \cdot 1\right)+\left(1-4^{2} \cdot 3\right)+\left(1-6^{2} \cdot 5\right)+\ldots \ldots+\left(1-20^{2} \cdot 19\right)$ $=\alpha-220 \beta,$ then an ordered pair $(\alpha, \beta)$ is equal to :

• A. (10,97)
• B. (11,103)
• C. (10,103)
• D. (11,97)

Answer 1

Answer: (B)

$1+\left(1-2^{2} \cdot 1\right)+\left(1-4^{2} \cdot 3\right)+\ldots \ldots+\left(1-20^{2} \cdot 19\right)$
$=\alpha-220 \beta$
$=11-\left(2^{2} \cdot 1+4^{2} \cdot 3+\ldots \ldots+20^{2} \cdot 19\right)$
$=11-2^{2} \cdot \displaystyle\sum_{ r =1}^{10} r ^{2}(2 r -1)=11-4\left(\frac{110^{2}}{2}-35 \times 11\right)$
$=11-220(103)$
$\Rightarrow \alpha=11, \beta=103$