# Math Help - I collected questions from different papers and I tried to solve them by myself and

1. ## I collected questions from different papers and I tried to solve them by myself and

Hi

I collected questions from different papers and I tried to solve them by myself and I want to correct them

2. Hello r-soy

(1) $d = 3$. Correct.

$S_{51} = \frac{51}{2}\Big(2(3)+50(\color{red}3\color{black} )\Big) = 3978$

(2) The algebra in the induction proof is not complete. You need to say:
$\tfrac13(4k^3-k)+(2(k+1)-1)^2 = \tfrac13(4k^3-k)+(2k+1)^2$
$=\tfrac13(4k^3-k+3[4k^2+4k+1])$

$=\tfrac13(4k^3+12k^2+11k+3)$

$=\tfrac13(4k^3+12k^2+12k+4-[k+1])$

$=\tfrac13(4[k+1]^3-[k+1])$
(3) Again, your working is not complete:
$\left(\frac{y}{x}\right)^{k+1}=\left(\frac{y}{x}\r ight)^{k}\cdot\frac{y}{x}$
$=\frac{y^k}{x^k}\cdot\frac yx$

$=\frac{y^{k+1}}{x^{k+1}}$