Finding the sum of an infinite series.

Question

Question:

If $$x = \frac{7}{4\times 1\times 2} + \frac{10}{4^2\times 2\times 3} + \frac{13}{4^3\times 3\times 4} +\dots $$ then find the value of $x$.

I managed to find the $n^\text{th}$ term as

$$t_n = \frac{\frac{4}{n} - \frac{1}{1+n}}{4^n}$$

I also tried to find some arithmetic or geometric series hidden in the question, but $1\times 2$, $2\times 3$,... in the denominator were quite problematic for me to deal with.

Please provide a solution which does not make use of calculus as I haven't taken it yet. Thank you.

Answer 1

Hint:your general term is $$\frac{4+3r}{4^r(r)(r+1)}=\frac{1}{4^{r-1}r}-\frac{1}{4^r(r+1)}$$

Now telescope.....

Answer 2

Hint. One classically has $$ \sum_{n=1}^\infty\frac{x^n}{n}=-\ln(1-x),\quad |x|<1, $$$$ \sum_{n=1}^\infty\frac{x^n}{n+1}=-1-\frac{\ln(1-x)}{x},\quad |x|<1. $$