Series：練習問題

Which of the following is the correct expression for the sum of the finite series \(\sum_{r=1}^n (4r - 3)\)?

Which of the following expressions is equivalent to the sum of the finite series \(\sum_{r=1}^n r^2(r+2)\)?

Using the standard result for \( \sum_{r=1}^n r \), state the value of \( \sum_{r=1}^{50} r \).

Evaluate the finite series \( \sum_{r=1}^{10} (r^2 + 1) \) using standard summation formulae.

Show that the sum of the series \( \sum_{r=1}^n (r^2 - r) \) can be expressed in the form \( \frac{1}{3}n(n+1)(n-1) \).

Given that \(\sum_{r=1}^{n} r = \frac{1}{2}n(n+1)\) and \(\sum_{r=1}^{n} r^2 = \frac{1}{6}n(n+1)(2n+1)\),
(a) Show that \(\sum_{r=1}^{n} (r^2 - r) = \frac{1}{3}n(n+1)(n-1)\).
(b) Hence, evaluate \(\sum_{r=10}^{20} (r^2 - r)\).