ok here we go.

**Question 1:**
$\displaystyle 1 + 2 + 3 +...+n = \frac{n(n+1)}{2}$

after a few lines of working.

assuming n=k is true.

prove for n=(k+1)

$\displaystyle \frac {k(k+1)}{2}+(k+1)=\frac {(k+1)(k+1+1)}{2}$

$\displaystyle (k+1)(k+2) = (k+1)(k+2)$

**Question 2:**
$\displaystyle 1 + 3 + 5+...+(2n-1) = n^2$

test for n = 1

$\displaystyle (2\times n +1)=n^2$

$\displaystyle 1 = 1$

assume true for n=k

prove for n=k+1

$\displaystyle k^2+(2(k+1)+1) = (k+1)^2$

$\displaystyle (k+1)^2= (k+1)^2$

**Question 3:**
$\displaystyle 1^2+2^2+3^2+...+n^2=\frac {n(n+1)(2n+1)}{6}$

test for n=1

$\displaystyle 1^2 = \frac {1 \times 2 \times 3}{6}$

$\displaystyle 1 = 1$

assume true for n=k

prove for n=k+1

$\displaystyle \frac {k(k+1)(2k+1)}{6} + (k+1)^2 = \frac {(k+1)(k+2)(2(k+1)+1)}{6}$

$\displaystyle LHS = RHS = 2k^3 + 9k^2 + 3k +6$

I think there Right

thanks for those if they are incorrect please point out. but I think I nailed them.

some more questions a little harder please.

~Regards

Name101