Question

Find the sum:

mc001-1.jpg

mc001-2.jpg

mc001-3.jpg

mc001-4.jpg

mc001-5.jpg

Answer 1

I can't view the image files you attached. Could you either

- paste the five sums (the math expressions) as text here, or
- upload the images directly, or
- type one of them and I can help with that first?

If helpful, here are quick formulas/methods you can use depending on the type of sum:

- Arithmetic: sum_{k=1}^n k = n(n+1)/2.
- Quadratic: sum_{k=1}^n k^2 = n(n+1)(2n+1)/6.
- Geometric: sum_{k=0}^{n} r^k = (1−r^{n+1})/(1−r) (r ≠ 1).
- Weighted geometric: sum_{k=0}^{n} k r^k = [r − (n+1)r^{n+1} + n r^{n+2}]/(1−r)^2.
- Telescoping: try partial-fraction decomposition to cancel terms.
- Infinite special sums: sum_{k=1}^\infty 1/k^2 = π^2/6; arctan series and Fourier expansions can produce constants like π.
- Use generating functions or differentiate/ integrate a known series if terms involve k, k^2, etc.

Paste one of the sums and I’ll solve it step by step.