If 1+(1-2^21)+(1-4^23)+(1-6^25)++(1-20^219) = -220, then the ordered pair (,) is equal to:

Question

If $1+(1-2^2\cdot1)+(1-4^2\cdot3)+(1-6^2\cdot5)+\cdots+(1-20^2\cdot19) = \alpha-220\beta$, then the ordered pair $(\alpha,\beta)$ is equal to:

Accepted Answer

$$S = 1+\sum_{r=1}^{10}\big[1-(2r)^2(2r-1)\big] = 11 - \sum_{r=1}^{10}(8r^3-4r^2) = 11 - \left[8(55)^2\cdot\frac{1}{?}\right]\cdots$$

Working through: $\sum 8r^3 = 2(110)^2$, $\sum4r^2=4(385)$, giving $S=11-220(110-7) = 11-220(103)$.

So $(\alpha,\beta)=(11,103)$.

Solutions · 1