If 1+(1--22.1)+(1--42.3)+(1-62.5)+......+(1-202.19)= [ ] - 220[ ], then an ordered pair [(…

Question

If 1+(1--22.1)+(1--42.3)+(1-62.5)+......+(1-202.19)=  $$\alpha $$ - 220$$\beta $$,
then an ordered pair $$\left( {\alpha ,\beta } \right)$$ is equal to:

Accepted Answer

$$1 + (1 - {2^2}.1) + (1 - {4^2}.3) + (1 - {6^2}.5) + ....(1 - {20^2}.19)$$

$$S = 1 + \sum\limits_{r = 1}^{10} {\left[ {1 - {{(2r)}^2}(2r - 1)} \right] = 1 + \sum\limits_{r = 1}^{10} {\left( {1 - 8{r^3} + 4{r^2}} \right)}  = 1 + 10 - } \sum\limits_{r = 1}^{10} {\left( {8{r^3} - 4{r^2}} \right)} $$

$$= 11 - 8{\left( {{{10 \times 11} \over 2}} \right)^2} + 4 \times \left( {{{10 \times 11 \times 21} \over 6}} \right)$$

$$ = 11 - 2 \times {(110)^2} + 4 \times 55 \times 7$$

$= 11 - 220(110 - 7)$

$$ = 11 - 220 \times 103 = \alpha  - 220\beta  $$

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

Solutions · 1