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 4 of 5: Divide by P

1-x^2R(x)=-\frac1{120}\frac{(x^2-1)(x^2-4)(x^2-9)(x^2-16)(x^2-25)}{(x^2+1)(x^2+2)(x^2+3)(x^2+4)(x^2+5)}

Detailed analysis

Divide the identity by P(x). Since Q(x)=R(x)P(x), this gives the displayed formula for 1−x^2R(x).