Find the coefficient of x in the expansion of ( (x)/(3) + \frac{9}{ {x}^(2) } ) ^(7) ​

Question

Find the coefficient of x in the expansion of


`( (x)/(3) + \frac{9}{ {x}^(2) } ) ^(7)`
​

Answer 1

By the binomial theorem,

`\left(\frac x3+\frac9{x^2}\right)^7=\displaystyle\sum_(n=0)^7\binom7n\left(\frac x3\right)^(7-n)\left(\frac9{x^2}\right)^7=\sum_(n=0)^7\binom7n\left(\frac3x\right)^(3n-7)`

We get the
`x` term when 3n - 7 = -1, or n = 2, which has coefficient

`\dbinom723^(3\cdot2-7)=\boxed{7}`