Consider the given factorial expression and evaluate it. 

21!/2!19!

Prove that the given statement is true for every positive integer n. Use mathematical induction.

5 + 25 + 125 + ... + 5ⁿ = (5^{n + 1} - 5)/4

Write S_k and S_{k + 1} for the given statement S_n where n is any positive integer. Simplify S_{k + 1} fully.

S_n: 2 is a factor of n² + 3n.

Rewrite the following sum using summation notation. Represent the index and lower limit of the summation as k and any number of your choice, respectively. Do not evaluate.

(p + q) + (p + 3q) + (p + 5q) + ⋯ + [p + (2n - 1)q]

Prove that the given statement is true for every positive integer n. Use mathematical induction.

1 · 4 + 2 · 5 + 3 · 8 + ... + n(n + 3) = n(n + 1)(n + 5)/3

Consider the given factorial expression and completely simplify it.

(n+4)!/(n+5)!

Find the following sum: