Problem 2

Let

a_1,a_2,a_3,a_4,a_5

be real numbers satisfying

\frac{a_1}{k^2+1}+\frac{a_2}{k^2+2}+\frac{a_3}{k^2+3}+\frac{a_4}{k^2+4}+\frac{a_5}{k^2+5}=\frac1{k^2}

for

k=1,2,3,4,5

. Find the value of

\frac{a_1}{37}+\frac{a_2}{38}+\frac{a_3}{39}+\frac{a_4}{40}+\frac{a_5}{41}

(express the value in a single fraction).

Step 2 of 5: Clear the denominators

P(x)=\prod_{j=1}^{5}(x^2+j),\quad Q(x)=R(x)P(x)

Detailed analysis

Let P(x)=(x^2+1)…(x^2+5) and Q(x)=R(x)P(x). At each x=±k, Q(k)=P(k)/k^2, so P(x)−x^2Q(x) vanishes at ±1,±2,±3,±4,±5.