Question

Find the value of $7$th term in the expansion of ${\left(3x^2+\frac{5}{2}y\right)}^{10}$.

Answer 1

Value of $7^{th}$ term of ${(3x^2+\frac{5}{2}y)}^{10}$

${(3x^2+\frac{5}{2}y)}^{10}=(3x^2{)}^{10}+{}^{10}{C}_1\times (3x^2{)}^9\times {\left(\frac{5}{2}y\right)}^1+...\dots$

$r^{th}$ term$={}^{10}{C}_{r-1}(3x^2{)}^{11-r}{\left(\frac{5}{2}y\right)}^{r-1}$

$7^{th}$ term $={}^{10}{C}_{7-1}(3x^2{)}^{(11-7)}{\left(\frac{5}{2}y\right)}^{(7-1)}$

$={}^{10}{C}_6\times 3^4{x}^8\times {\left(\frac{5}{2}\right)}^6{y}^6$

$=\frac{10\times 9\times 8\times 7}{4\times 3\times 2}\times 3^4\times \frac{5^6}{2^6}{x}^8{y}^6$

$7^{th}$ term$=\frac{3^5\times 5^7\times 7}{2^5}{x}^8{y}^6$.