Recurrence formula

We want to find the value $\displaystyle\sum\limits_{k=1}^{10} 5^{k}$ . We are given that the value at $k = 5$ is $4$ . We wish to find the values at $25$ , $125$ , $625$ , etc, all the way up to $5^{10}$ . We are given that with every factor of $5$ that we increase the value by, we add $3$ to the value. We are increasing by a factor of $5$ with each iteration of the summation. Therefore, $f(25)=7, f(125)=10, f(625)=13, f(3125)=16, \cdots, f(5^{10})=31$ . We must find the sum $4+7+10+13+16+19+22+25+28+31$ , which is $\fbox{175}$

f(25)=f(5)+3=4+3=7 f(125)=f(25)+3=7+3 therefore f(5),f(25),f(125),f(625)... forms an AP with first term 4 and c.d 3