Find the first three partial sums and the nth partial sum of the sequence given by an=1/n+1-1/n+2
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.
To find the partial sums of the sequence given by an = 1/(n + 1) - 1/(n + 2), we can start by finding the first few terms:
a1 = 1/(1 + 1) - 1/(1 + 2) = 1/2 - 1/3 = 1/6
a2 = 1/(2 + 1) - 1/(2 + 2) = 1/3 - 1/4 = 1/12
a3 = 1/(3 + 1) - 1/(3 + 2) = 1/4 - 1/5 = 1/20
The partial sums can be obtained by adding the first n terms:
S1 = a1 = 1/6
S2 = a1 + a2 = 1/6 + 1/12 = 1/4
S3 = a1 + a2 + a3 = 1/6 + 1/12 + 1/20 = 5/12
To find the nth partial sum (Sn), we can generalize the pattern observed in the partial sums:
Sn = 1/6 + 1/12 + 1/20 + ... + 1/(n + 1) - 1/(n + 2)
We can simplify this by finding a common denominator for all the terms:
Sn = (n + 2) / ((n + 1)(n + 2)) + (n + 1) / ((n + 1)(n + 2)) + ... + 1/(n + 1) - 1/(n + 2)
Sn = ((n + 2) + (n + 1) + ... + 1) / ((n + 1)(n + 2)) - 1/(n + 2)
Sn = ((n + 2)(n + 1)/2) / ((n + 1)(n + 2)) - 1/(n + 2)
Sn = 1/2 - 1/(n + 2)
So, the nth partial sum of the sequence is Sn = 1/2 - 1/(n + 2).