Question

Statement 1 : Coefficient of $x^{14}$ in ${(1+2x+{3x}^2\cdots +{16x}^{15})}^2$ is 560
Statement 2 : ${\sum }_{r=1}^{n}r(n-r)=\frac{n(n^2-1)}{6}$

• A. Both statements are true and statement 2 is the correct explanation of statement 1
• B. Both statements are true but statement 2 is not the correct explanation of statement 1
• C. Statement 1 is true and statement 2 is false
• D. Statement 1 is false and statement 2 is true

Answer 1

The correct option is A Both statements are true and statement 2 is the correct explanation of statement 1

The coefficient of $x^{14}$ is $1\ast 15+2\ast 14+3\ast 13+....15\ast 1$
This is equal to the second expression where $n=15$
Statement 2: ${\sum }_{r=1}^{n}r(n-r)=\frac{n(n^2-1)}{6}$
Consider,${\sum }_{r=1}^{n}r(n-r)=1(n-1)+2(n-2)+3(n-3)+...+n(n-n)$
$=(n+2n+...n(n\left)\right)-(1+2^2+3^2+...+n^2)$
$=n^2\frac{(n+1)}{2}-\frac{n(n+1)(2n+1)}{6}$
$=\frac{n(n+1)}{2}[n-\frac{(2n+1)}{3}]$
$=\frac{n(n^2-1)}{6}$
Hence, statement 2 is true and explains statement 1.