+218 Compute \(\frac{1}{1 \times 4} + \frac{1}{4 \times 7} + \frac{1}{7 \times 10} + \dots + \frac{1}{37 \times 40}.\)
+128707 Using partial fraction decomposition :
1/ [n * (n + 3)] = A / n + B / (n +3) multiply through by n *(n + 3)
1 = A (n + 3) + Bn
1 = (A + B)n + 3A
3A = 1
A = 1/3
A+B = 0
B = -1/3
So
1 / [1x 4] = (1/3)/1 - (1/3)/4 = 1/3 - 1/12
1/ [ 4 x 7] = (1/3)/4 - (1/3)/7 = 1/12 - 1/21
1/[7 * 10] = (1/3)/7 - (1/3)/10 = 1/21 - 1/30
......
1/(34 *37) = ((1/3)/34 - (1/3)/37 = 1/ 102 - 1/111
1 / (37 * 40) = (1/3)/(37) - (1/3)/40 = 1/111 - 1/120
All the intermediate terms "cancel" and we are left with
1/3 - 1/120 =
[ 120 - 3 ] / 360 =
117 / 360 =
13 / 40
