Suppose that the nth partial sum of the series
∞
Σan is Sn = (5n-2)/(5n+2)
n=1
Find a3. I have tried plugging in n = 3 and solving Sn but that is incorrect. I believe something is wrong with my approach and I would be happy to get some guidance.
S3-S2 = term 3
S3 = 13/17
S2 = 8/12
maybe 13/17 - 8/12 ??? about .098
To find the value of a3, we need to use the formula for the nth partial sum and solve for a3.
The formula for the nth partial sum is given by:
Sn = (5n - 2) / (5n + 2)
To find a3, we can plug in n = 3 into the formula:
S3 = (5(3) - 2) / (5(3) + 2)
Simplifying this expression gives us:
S3 = (15 - 2) / (15 + 2)
S3 = 13 / 17
Therefore, a3 = 13.
To find the value of a₃, we need to evaluate the given series sum formula.
Let's start by examining the given series sum formula:
Sₙ = (5n - 2) / (5n + 2)
The subscript, n, represents the term numbers of the series. In our case, we want to find the third term, so n = 3.
Now let's substitute n = 3 into the series sum formula:
S₃ = (5(3) - 2) / (5(3) + 2)
= (15 - 2) / (15 + 2)
= 13 / 17
Therefore, the value of the third partial sum, S₃, is 13/17.
However, it seems there might be some confusion, as the term a₃ represents the value of the third term of the given series (not the partial sum). To find a₃, we can find the difference between consecutive partial sums.
a₃ = S₃ - S₂
To find S₂, we can substitute n = 2 into the series sum formula:
S₂ = (5(2) - 2) / (5(2) + 2)
= 8 / 12
= 2 / 3
Now, let's calculate the value of a₃:
a₃ = S₃ - S₂
= (13/17) - (2/3)
= (39 - 34) / (51)
= 5 / 51
Therefore, the value of a₃ is 5/51.
To recap:
- The value of the third partial sum, S₃, is 13/17.
- The value of the third term, a₃, is 5/51.