Question

What is the sum of the coefficient of the two middle terms ?

• A. 0
• B. 1
• C. -1
• D. None of the above

Answer 1

The correct option is A 0

Using binomial theorem, we can write

${(x^3-\frac{1}{x^2})}^n=\sum _{r=0}^n{}_r^{n}C{{(x}^3)}^{n-r}{\left(\frac{-1}{x^2}\right)}^r=\sum _{r=0}^n{(-1)}^r{}_r^{n}C{x}^{(3n-5r)}$

Number of terms in the expansion $=n+1=16$

Middle terms are $8^{th}$ and $9^{th}$

For which $r=7$ and $8$

Sum of coefficients $={(-1)}^7{}_7^{15}C+$${(-1)}^8{}_8^{15}C$

Using the property of combinations, we have

${}_7^{15}C-$${}_8^{15}C={}_8^{15}C-{}_8^{15}C=0$

Hence, sum of the coefficients is $0$.