If S = 75+95^2+135^3+195^4+, then 160S is equal to ________.

Question

If $S = \dfrac75+\dfrac{9}{5^2}+\dfrac{13}{5^3}+\dfrac{19}{5^4}+\cdots$, then $160S$ is equal to \_\_\_\_\_\_\_\_.

Answer: 305

Accepted Answer

The numerators satisfy $T_n = n^2-n+7$ (check: $T_1=7,T_2=9,T_3=13,T_4=19$).

Using $\sum n^2x^n = \dfrac{x(1+x)}{(1-x)^3}$, $\sum nx^n=\dfrac{x}{(1-x)^2}$ with $x=\tfrac15$:

$$S = \frac{15}{32}-\frac{5}{16}+\frac74 = \frac{15-10+56}{32} = \frac{61}{32}.$$

$$160S = 160\times\frac{61}{32} = 5\times61 = 305.$$

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