mathematical induction.

$($a$)$ Use $P(k)$ to show $P(k+1).$

$2+4+6+$cdots$+2n=n(n+1)$

$($b$)$ Is $P(n)$ true for all $n1?$

$1^2+3^2+5^2+$cdots$+(2n-1{)}^2=\frac{n(2n+1)(2n-1)}{3}$

Let $P(n)$:$1+5+9+$cdots$+(4n-3)=(2n+1)(n-1).$

$1+2^1+2^2+$cdots$+2^n=2^{n+1}-1$

$($a$)$ Use $P(k)$ to show $P(k+1).$

$5+10+15+$cdots$+5n=\frac{5n(n+1)}{2}$

$($b$)$ Is $P(n)$ true for all $n1?$

Prove $1+2^n<msup\; id="EditorElement\_158928"><mrow><mn\; id="EditorElement\_158921">3</mn></mrow><mrow\; id="EditorElement\_158927"><mi\; id="EditorElement\_158924">n</mi></mrow></msup>$ for $n2.$

$1^2+2^2+3^2+$cdots$+n^2=\frac{n(n+1)(2n+1)}{6}$

Prove $n<msup\; id="EditorElement\_158699"><mrow><mn\; id="EditorElement\_158692">2</mn></mrow><mrow\; id="EditorElement\_158698"><mi\; id="EditorElement\_158695">n</mi></mrow></msup>$ for $n>1.$

Prove $1+2+3+$cdots

$1+a+a^2+$cdots$+a^{n-1}=\frac{a^n-1}{a-1}$

Find the least $n$ for which the statement is true and then prove that

Find the least $n$ for which the statement is true and then