Question

Find the $24th$ term in the sequence whose nth term $a_n=\frac{n\left(n-2\right)}{n+3}$ . using binomial theorem , Evaluate ${\left(102\right)}^3$

Answer 1

$a_{24}=\left(\frac{24(24-2)}{24+3}\right)$

$=\left(\frac{24\cdot 22}{27}\right)$

$=\left(\frac{176}{9}\right)$

$Now(102{)}^3=(100+2{)}^3$

$={}^3{C}_0(100{)}^3\cdot 2^0+{}^3{C}_1(100{)}^2\cdot 2^1+{}^3{C}_2(100{)}^1\cdot 2^2+{}^3{C}_3(100{)}^0\cdot 2^3$

$=1000000+60000+1200+8$

$=1061208$