<s>[INST] Sum of a series of a number raised to incrementing powers-- for any number Sum of a series of a number raised to incrementing powers. I have a sub question on this. As I dont have reputation to ask for in comment. The formula being derived for series of summation of powers to 2 i.e $2^{n+1}-1$ doesnt apply for other numbers suppose $5^0+5^1+5^2+5^3+5^4$ is not equal to $5^{4+1}-1$. Hence I request if there is any generalized formula i.e that will give the summation of series of powers of the number available.
If any such formula is available need your help!!
 [/INST] This is a geometric series
$$S=5^0+5^1+5^2+5^3+5^4+\cdots+5^n$$
$$5S=5^1+5^2+5^3+5^4+5^5+\cdots+5^{n+1}$$
$$4S=5^{n+1}-1$$
$$S=\frac{5^{n+1}-1}{4}$$
In general to find $1+x+x^2+\cdots+x^n$,
$$\sum_{j=0}^{n}x^j=\frac{x^{n+1}-1}{x-1}$$
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