Mathematics - Sequence & Series Question with Solution | TestHub

Item A: the coefficient of $x^{49}$ is $-(1+3+5+\cdots +99)=-50^2=-2500$. A matches p. Item B: $\frac{S_{2n}}{S_n}=3$ gives $d=\frac{2a}{n+1}$. Then $\frac{S_{3n}}{S_n}=\frac{3[2a+(3n-1)d]}{2a+(n-1)d}=\frac{3[(n+1)+(3n-1)]}{(n+1)+(n-1)}=\frac{12n}{2n}=6$. B matches s. Item C: $\frac{1}{2}{\sum }_{r=2}^{\infty }\left(\frac{1}{r-1}-\frac{1}{r+1}\right)=\frac{1}{2}\left(1+\frac{1}{2}\right)=\frac{3}{4}$. C matches r. Item D: Let the dimensions be $l,lr,l{r}^2$ (taking $l$ as the largest, so $r<1$ and the product is $l^3{r}^3$ ). With $l^3{r}^3=27$, so $lr=3$. Surface area $2\left(l^{2}r+l^2{r}^3+l^2{r}^2\right)=78$ gives $l^2\left(r+r^2+r^3\right)=39$, so $l+3+\frac{9}{l}=13,l^2-10l+9=0,l=1$ or 9 . Since $l$ is the largest, $r=\frac{1}{3}$ and $l=9$. D matches q.