Question Type 3: Verifying the proof Exercises

Verify that for $m=3$, the identity $1^2+2^2+\cdots+m^2=\frac{m(m+1)(2m+1)}{6}$ holds.

Verify that for $m=3$, the sum of cubes satisfies $1^3+2^3+\cdots+m^3=\left(\frac{m(m+1)}{2}\right)^2$.

Verify that for $m=3$, the expression $m^2-m$ is even.

Verify for $m=3$ that the sum of the first $m$ odd numbers equals $m^2$.

Verify that for $n=3$, the inequality $n!\ge2^{n-1}$ holds.

Expand and verify the binomial theorem for $n=3$ in the expansion of $(a+b)^n$.

Verify that for $m=3$, the geometric sum $2^0+2^1+\dots+2^m=2^{m+1}-1$ holds.

Verify for $m=3$ that $\displaystyle\prod_{k=1}^m\Bigl(1+\frac1k\Bigr)=m+1$.

Verify the base case $m=3$ for the formula $1+2+\dots+m=\frac{m(m+1)}{2}$.

Verify that for $m=3$, $3$ divides $m^3-m$.

Verify for $n=3$ that $5^n - 4n - 1$ is divisible by $8$.