Find the first three partial sums and the nth partial sums of the sequence

Question

An=1/n+2-1/n+3

Answer 1

The sequence is represented by the formula An = 1/(n+2) - 1/(n+3).

To find the first three partial sums, we need to add up the terms of the sequence:

First partial sum:
A1 = 1/(1+2) - 1/(1+3) = 1/3 - 1/4 = 4/12 - 3/12 = 1/12

Second partial sum:
A1 + A2 = (1/3) + 1/(3+2) - 1/(3+3) = 1/3 + 1/5 - 1/6 = 10/30 + 6/30 - 5/30 = 11/30

Third partial sum:
A1 + A2 + A3 = (1/3) + (1/5) + 1/(5+2) - 1/(5+3) = 1/3 + 1/5 + 1/7 - 1/8 = 280/840 + 504/840 + 360/840 - 315/840 = 829/840

To find the nth partial sum, we can use a general formula:

nth partial sum = A1 + A2 + A3 + ... + An

With the given sequence An = 1/(n+2) - 1/(n+3), it is not possible to find a simple closed-form expression for the nth partial sum.