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}`