Question

Sum of first n terms of an $AP$ is given by $S=\frac{5n^2}{\alpha }+\frac{3n}{\alpha }$ Find the $20th$ term of the $A.P.$

Answer 1

$t_n=S_n-S_{n-1}$

$=\frac{5n^2}{\alpha }+\frac{3n}{\alpha }-\frac{5{(n-1)}^2}{\alpha }-\frac{3(n-1)}{\alpha }$

$=\frac{1}{\alpha }[5(n^2-{(n-1)}^2)+2(n-n+1)]$

$=\frac{1}{\alpha }[5(n-n+1)(n+n-1)+2\left(1\right)]$

$=\frac{1}{\alpha }[5(2n-1)+2]$

$=\frac{1}{\alpha }[10n-5+2]$

$\therefore t_n=\frac{10n-3}{\alpha }$

${20}^{th}$ term$=t_{20}=\frac{10\times 20-3}{\alpha }=\frac{200-3}{\alpha }=\frac{197}{\alpha }$

$\therefore {20}^{th}$ term is $\frac{197}{\alpha }$