Find the first three partial sums and the nth partial sum of the sequence given by an=1/n+1-1/n+2

Answer 1

To find the partial sum, we need to find the sum of the terms in the sequence up to a given value of n.

Let's find the first three partial sums:
1. For n = 1, the first term is a₁ = 1/1+1 - 1/1+2 = 1/2 - 1/3 = (3 - 2)/6 = 1/6.
2. For n = 2, the second term is a₂ = 1/2+1 - 1/2+2 = 1/3 - 1/4 = (4 - 3)/12 = 1/12. To find the partial sum up to the second term, we need to add the first and second terms: partial sum = a₁ + a₂ = 1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4.
3. For n = 3, the third term is a₃ = 1/3+1 - 1/3+2 = 1/4 - 1/5 = (5 - 4)/20 = 1/20. To find the partial sum up to the third term, we need to add the first, second, and third terms: partial sum = a₁ + a₂ + a₃ = 1/6 + 1/12 + 1/20.

Now, let's find the nth partial sum:
To find the nth partial sum, we need to add up all the terms from the first term to the nth term:
n-th partial sum = a₁ + a₂ + a₃ + ... + aₙ

The general term for the sequence is aₙ = 1/(n+1) - 1/(n+2).
To find the nth partial sum, we can write out the sum of all the terms up to the nth term:
n-th partial sum = (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) + ... + (1/n+1 - 1/n+2).

However, notice that most of the terms cancel out when we write it this way. Let's simplify the expression:
n-th partial sum = (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) + ... + (1/n+1 - 1/n+2)
= 1/2 - 1/n+2

Therefore, the nth partial sum of the sequence is given by the formula Sₙ = 1/2 - 1/n+2.