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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!!

Naseer Mohammad
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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 $1+x+x^2+\cdots+x^n$, $$\sum_{j=0}^{n}x^j=\frac{x^{n+1}-1}{x-1}$$

Cheong Sik Feng
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