If <math><mn>1</mn><mo>+</mo><mfenced separators="|"><mrow><mrow><mn>1</mn><mo>-</mo><msup><mrow><mrow><mn>2</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>⋅</mo><mn>1</mn></mrow></mrow></mfenced><mo>+</mo><mfenced separators="|"><mrow><mrow><mn>1</mn><mo>-</mo><msup><mrow><mrow><mn>4</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>⋅</mo><mn>3</mn></mrow></mrow></mfenced><mo>+</mo><mfenced separators="|"><mrow><mrow><mn>1</mn><mo>-</mo><msup><mrow><mrow><mn>6</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>⋅</mo><mn>5</mn></mrow></mrow></mfenced><mo>+</mo><mo>…</mo><mo>…</mo><mo>.</mo><mo>+</mo><mfenced separators="|"><mrow><mrow><mn>1</mn><mo>-</mo><msup><mrow><mrow><mn>20</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>⋅</mo><mn>19</mn></mrow></mrow></mfenced><mo>=</mo><mi>α</mi><mo>-</mo><mn>220</mn><mi>β</mi></math> , then an ordered pair <math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math> is equal to:

1+1-22.1+1-42.3+…….+1-202.19=α-220β=11-22⋅1+42⋅3+…..+202⋅19=11-22⋅∑r=110? r2 (2r-1)=11-411022-35×11=11-220 (103)⇒α=11, β=103

If&nbsp;<math><mn>1</mn><mo>+</mo><mfenced separators="|"><mrow><mrow><mn>1</mn><mo>-</mo><msup><mrow><mrow><mn>2</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>⋅</mo><mn>1</mn></mrow></mrow></mfenced><mo>+</mo><mfenced separators="|"><mrow><mrow><mn>1</mn><mo>-</mo><msup><mrow><mrow><mn>4</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>⋅</mo><mn>3</mn></mrow></mrow></mfenced><mo>+</mo><mfenced separators="|"><mrow><mrow><mn>1</mn><mo>-</mo><msup><mrow><mrow><mn>6</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>⋅</mo><mn>5</mn></mrow></mrow></mfenced><mo>+</mo><mo>…</mo><mo>…</mo><mo>.</mo><mo>+</mo><mfenced separators="|"><mrow><mrow><mn>1</mn><mo>-</mo><msup><mrow><mrow><mn>20</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>⋅</mo><mn>19</mn></mrow></mrow></mfenced><mo>=</mo><mi>α</mi><mo>-</mo><mn>220</mn><mi>β</mi></math>&nbsp;, then an ordered pair&nbsp;<math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math>&nbsp;is equal to:

Ask & Answer Question

Option 1 -

$(11,97)$

Option 2 -

$(10,97)$

Option 3 -

$(10,103)$

Option 4 -

$(11,103)$

4 Views | Posted 2 months ago

Answered by

alok kumar singh | Contributor-Level 10

2 months ago

Correct Option - 4

Detailed Solution:

$1 + (1 - 2^{2} . 1) + (1 - 4^{2} . 3) + \dots \dots . + (1 - 20^{2} . 19)$

$= α - 220 β$

$= 11 - (2^{2} \cdot 1 + 4^{2} \cdot 3 + \dots . . + 20^{2} \cdot 19)$

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

$= 11 - 220 (103)$

$\Rightarrow α = 11, β = 103$

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