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 as well as summing the first three terms of the series to find a3. I believe something is wrong with my approach and I would be happy to get some guidance.
To find the value of a3, we need to first understand the formula for the nth partial sum of the series Sn. In this case, we have Sn = (5n-2)/(5n+2) for n = 1 onwards.
The formula for the nth term of a series is given by the difference between consecutive partial sums, which can be expressed as an = Sn - Sn-1.
To find a3, we need to calculate the value of S3 - S2. Let's break it down step by step:
1. Start by calculating S3:
S3 = (5(3)-2)/(5(3)+2)
= (15-2)/(15+2)
= 13/17
2. Next, calculate S2:
S2 = (5(2)-2)/(5(2)+2)
= (10-2)/(10+2)
= 8/12
= 2/3
3. Finally, we can find a3 by subtracting S2 from S3:
a3 = S3 - S2
= 13/17 - 2/3
To perform the subtraction, we need to find a common denominator:
a3 = (13/17)(3/3) - (2/3)(17/17)
= 39/51 - 34/51
= 5/51
Therefore, the value of a3 is 5/51.
In summary, to find a3, we calculate the difference between the third and second partial sums (S3 - S2) using the given formula.