Sum of 5 fractions
Source: APMO 2009 Q.2
March 13, 2009
linear algebraalgebra
Problem Statement
Let , , , , be real numbers satisfying the following equations:
\frac{a_1}{k^2\plus{}1}\plus{}\frac{a_2}{k^2\plus{}2}\plus{}\frac{a_3}{k^2\plus{}3}\plus{}\frac{a_4}{k^2\plus{}4}\plus{}\frac{a_5}{k^2\plus{}5} \equal{} \frac{1}{k^2} for k \equal{} 1, 2, 3, 4, 5
Find the value of \frac{a_1}{37}\plus{}\frac{a_2}{38}\plus{}\frac{a_3}{39}\plus{}\frac{a_4}{40}\plus{}\frac{a_5}{41} (Express the value in a single fraction.)