Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.
$\frac{n}{6}(n+1)(2n+25)$
1 \cdot 9 + 2 \cdot 10 + 3 \cdot 11 + \dots + n(n+8) =
According to the Principle of Mathematical Induction, what two conditions must the given statement satisfy to prove that it is true for all natural numbers? Select all that apply.
The statement is true for the natural number 1.
The statement is true for any two natural numbers k and k + 1.
If the statement is true for the natural number 1, it is also true for the next natural number 2.
If the statement is true for some natural number k, it is also true for the next natural number k + 1.
Show that the first of these conditions is satisfied by evaluating the left and right sides of the given statement for the first natural number.
1 \cdot 9 + 2 \cdot 10 + 3 \cdot 11 + \dots + n(n+8) = \frac{n}{6}(n+1)(2n+25)
=
(Simplify your answers.)
