I'm a beginner at evaluating summations of this level, and I would appreciate it if you could walk me through the steps of evaluating this type of problem. I'm having trouble understanding what I'm trying to reach or solve for and how to get there.
I didn't get very far in my attempt, and I don't know if it's right, but what I have so far is
$(x+1)^2 +(x+1)^3 +(x+1)^4+ … +(x+1)^n.$
Then you could take that and multiply by (x+1)
$(x+1)^3 + (x+1)^4 + … + (x+1)^n + (x+1)^{n+1}$
and then take the difference of the summations
$(x+1)^2 – (x+1)^{n+1}$
and I got stuck here — no idea on how to progress.
Thanks!
Best Answer
Let $$S_n=\sum_{k=1}^n (x+1)^{k+1}\tag1.$$ Multiplying $(1)$ by $(x+1)$, we get $$(x+1)S_n=\sum_{k=1}^n (x+1)^{k+2}\tag2.$$ $(2)-(1)\implies$ $$(x+1)S_n-S_n=\sum_{k=1}^n \left ( (x+1)^{k+2}-(x+1)^{k+1}\right).$$ Most of the terms from RHS will cancel out and we are left with $$x\cdot S_n=(x+1)^{n+2}-(x+1)^2.$$ In other words, $$S_n=\frac {(x+1)^2((x+1)^{n}-1)}{x}.$$ If you want, you can still simplify the expression using binomial expansion of $(x+1)^{n}$.