Question Type 4: Using induction for proving equalities Bootcamps

Prove by induction that the sum of the first $n$ odd numbers is $1 + 3 + 5 + \dots + (2n-1) = n^2$.

Consider the sequence defined by $a_1=2$ and $a_n=a_{n-1}+3\quad\text{for }n\ge2\text{.}$ Prove by induction that $a_n=3n-1\quad\text{for all }n\ge1\text{.}$

Prove by induction that $1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$.

Prove by induction that $1^3 + 2^3 + \dots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$.

Verify by induction that $1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}$ for all positive integers $n$.