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STD 11 - 8. sequence and series — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 8. sequence and series4 Marks
MCQ
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
    $(10,97)$
  • ✓
    $(11,103)$
  • C
    $(10,103)$
  • D
    $(11,97)$
✓

Answer

Correct option: B.
$(11,103)$
b
$1+\left(1-2^{2} \cdot 1\right)+\left(1-4^{2} \cdot 3\right)+\ldots \ldots+\left(1-20^{2} \cdot 19\right)$

$=\alpha-220 \beta$

$=11-\left(2^{2} \cdot 1+4^{2} \cdot 3+\ldots \ldots+20^{2} \cdot 19\right)$

$=11-2^{2} \cdot \sum_{ r =1}^{10} r ^{2}(2 r -1)=11-4\left(\frac{110^{2}}{2}-35 \times 11\right)$

$=11-220(103)$

$\Rightarrow \alpha=11, \beta=103$

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