Question

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.

Answer 1

[AI-generated answer — Explain Bot]

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.