I got a question which says
$$ 1 + \frac {2}{7} + \frac{4}{7^2} + \frac{7}{7^3} + \frac{11}{7^4} + \cdots$$
I got the solution by dividing by $7$ and subtracting it from original sum. Repeated for two times.(Suggest me if any other better way of doing this).
However now i am interested in understanding the series 1,2,4,7,11,….. In which the difference of the numbers are consecutive natural numbers.
How to find the sum of
$1+2+4+7+11+\cdots nterms$
This is my first question in MSE. If there are some guidelines i need to follow, which i am not, please let me know.
Best Answer
Your series (without the denominators) is $$\sum_{i=1}^n 1+ \frac {i(i-1)}2=\sum_{i=1}^n 1+\frac {i^2}2 - \frac i2=n+\frac {n(n+1)(2n+1)}{12}-\frac{n(n+1)}4$$ This is an application of Faulhaber's formula