HMMT-Nov 2009 gen, Problem 1

Evaluate the sum: \[ 11^{2}-1^{2}+12^{2}-2^{2}+13^{2}-3^{2}+\ldots+20^{2}-10^{2} \]

Solution: This sum can be written as $\sum_{a=1}^{10}(a+10)^{2}-a^{2}=\sum_{a=1}^{10} 10(2 a+10)=10 * 10 *$ $11+10 * 10 * 10=2100$. $\fbox{2100}$.

Answer: 2100