Inductive Proofs for Sums of Integers
Interactive practice questions
We want to find out if the equation $6+12+18+\dots+6n=3n^2+3n$6+12+18+…+6n=3n2+3n holds when $n=4$n=4.
Write the equation when $n=4$n=4.
Do not simplify each side.
Evaluate the left hand side of the equation.
Evaluate the right hand side of the equation.
Is the equation true or false when $n=4$n=4?
False
True
Suppose $S_n$Sn represents the following statement.
$9+18+27+\text{. . .}+9n=\frac{9n\left(n+1\right)}{2}$9+18+27+. . .+9n=9n(n+1)2
We want to find out if the equation $1^2+2^2+3^2+\dots+n^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6}$12+22+32+…+n2=n(n+1)(2n+1)6 holds when $n=4$n=4.
We want to find out if the equation $\frac{1}{2^1}+\frac{1}{2^2}+\frac{1}{2^3}+\dots+\frac{1}{2^n}=\frac{2^n-1}{2^n}$121+122+123+…+12n=2n−12n holds when $n=4$n=4.