Question

given the series 1+2+3+4+5+6+...+5000. Write the series in sigma notation if all the powers of 4 are removed from the series.​

Answer 1

We have 4⁶ = 4096 and 4⁷ = 16,384, which is to say that the given sum only contains the first six powers of 4.

Now,

$\displaystyle 1+2+3+\cdots+5000 = \sum_{k=1}^{5000}k$

and you subtract the sum of the first six powers of 4 to get the sum S that you want,

$\displaystyle S = \boxed{\sum_{k=1}^{5000}k - \sum_{k=1}^64^k}$