A Typical Recurrence Relation

From the recurrence relation $a_{n+2} = 2a_{n+1} + a_n$ , we have the characteristic equation $x^2 - 2x - 1 = 0$ and $x = 1 \pm \sqrt{2}$ . Therefore, we have:

$\begin{aligned} a_n & = c_1(1+\sqrt{2})^n + c_2(1-\sqrt{2})^n \\ a_0 = 1: \quad 1 & = c_1 + c_2 \\ a_1 = 1: \quad 1 & = c_1(1+\sqrt{2}) + c_2(1-\sqrt{2}) \\ 1 & = c_1 + c_2 + \sqrt{2}(c_1-c_2) \\ 1 & = 1 + \sqrt{2}(c_1-c_2) \\ \Rightarrow c_1 - c_2 & = 0 \\ c_1 = c_2 & = \frac{1}{2} \\ \Rightarrow a_n & = \frac{(1+\sqrt{2})^n + (1-\sqrt{2})^n}{2} \end{aligned}$

Now we have:

$\begin{aligned} S & = \sum_{n=0}^\infty \frac{a_n}{5^{2n}} \\ & = \frac{1}{2} \left(\sum_{n=0}^\infty \left(\frac{1+\sqrt{2}}{25}\right)^n + \sum_{n=0}^\infty \left(\frac{1-\sqrt{2}}{25}\right)^n \right) \\ & = \frac{1}{2} \left( \frac{1}{1 - \frac{1+\sqrt{2}}{25}} + \frac{1}{1 - \frac{1-\sqrt{2}}{25}} \right) \\ & = \frac{1}{2} \left( \frac{25}{24-\sqrt{2}} +\frac{25}{24+\sqrt{2}} \right) \\ & = \frac{1}{2} \left( \frac{2 \cdot{} 25 \cdot{} 24}{24^2 - 2} \right) \\ & = \frac{300}{287} \end{aligned}$

$\Rightarrow p+q = 300 + 287 = \boxed{587}$

$\quad S=\frac { 1 }{ { 25 }^{ 0 } } +\frac { 1 }{ { 25 }^{ 1 } } +\frac { 3 }{ { 25 }^{ 2 } } +\frac { 7 }{ { 25 }^{ 3 } } +\frac { { 17 } }{ { 25 }^{ 4 } } +\frac { 41 }{ { 25 }^{ 5 } } +\frac { 99 }{ { 25 }^{ 6 } } ......\\ \\ \frac { S }{ { 25 }^{ 2 } } =\frac { 1 }{ { 25 }^{ 2 } } +\frac { 1 }{ { 25 }^{ 3 } } +\frac { 3 }{ { 25 }^{ 4 } } +\frac { 7 }{ { 25 }^{ 5 } } +\frac { { 17 } }{ { 25 }^{ 6 } } +\frac { 41 }{ { 25 }^{ 7 } } +\frac { 99 }{ { 25 }^{ 8 } } ......\\ \\ \therefore \frac { 624S }{ 625 } =\frac { 1 }{ { 25 }^{ 0 } } +\frac { 1 }{ { 25 }^{ 1 } } +2\left( \frac { 1 }{ { 25 }^{ 2 } } +\frac { 3 }{ { 25 }^{ 3 } } +\frac { 7 }{ { 25 }^{ 4 } } +\frac { 17 }{ { 25 }^{ 5 } } +\frac { { 41 } }{ { 25 }^{ 6 } } ...... \right) \\ \\ \Rightarrow \therefore \frac { 624S }{ 625 } =\frac { 1 }{ { 25 }^{ 0 } } +\frac { 1 }{ { 25 }^{ 1 } } +\frac { 2 }{ 25 } \left( \frac { 1 }{ { 25 }^{ 1 } } +\frac { 3 }{ { 25 }^{ 2 } } +\frac { 7 }{ { 25 }^{ 3 } } +\frac { 17 }{ { 25 }^{ 4 } } +\frac { { 41 } }{ { 25 }^{ 5 } } ....... \right) \\ \\ \Rightarrow \frac { 624S }{ 625 } =\frac { 1 }{ { 25 }^{ 0 } } +\frac { 1 }{ { 25 }^{ 1 } } +\frac { 2 }{ 25 } \left( S-1 \right) \\ \\ \Rightarrow 574S=600\\ \\ \Rightarrow S=\frac { 300 }{ 287 } =\frac { P }{ Q } \\ \\ \therefore \quad \boxed { P+Q=587 } \\ \\$