The general term of a sequence can be found from its partial sums by using the relationship between the partial sums and the sequence itself.
Here's a breakdown of the process:
Understanding Partial Sums
A partial sum, denoted as Sn, represents the sum of the first n terms of a sequence. For instance, as mentioned in the reference, S3 (the third partial sum) is a1 + a2 + a3, where a1, a2, and a3 are the first three terms of the sequence. In general:
Sn = a1 + a2 + a3 + ... + an
Finding the General Term (an)
The core idea to find the general term an from Sn is that an is the difference between two consecutive partial sums, that is Sn and Sn-1.
Here's how to find the general term (an) from a sequence of partial sums (Sn):
- Use the relationship: an = Sn - Sn-1 for n > 1. This formula expresses the nth term as the difference between the sum of the first n terms and the sum of the first n-1 terms.
- Calculate a1: The first term (a1) is always equal to the first partial sum, i.e., a1 = S1.
- Apply and simplify: Substitute the expression for Sn and Sn-1 into the formula and simplify to obtain a general expression for an.
Example
Let's say Sn = n2 + n. Here's how you would find the general term:
- Find a1: a1 = S1 = 12 + 1 = 2
- Find Sn-1: Sn-1 = (n-1)2 + (n-1) = n2 - 2n + 1 + n - 1 = n2 - n
- Use the relationship an = Sn - Sn-1: an = (n2 + n) - (n2 - n) = 2n for n > 1.
Therefore, the general term is:
- a1 = 2
- an = 2n for n > 1
Note that in some cases you may need to verify if the an formula also holds true for n=1. In our example, 2*1 = 2, which is equal to a1. So the general term for this sequence is an = 2n.
Summary
| Step | Description |
|---|---|
| 1 | Calculate a1 using a1 = S1. |
| 2 | Calculate Sn-1 by replacing n with n-1 in the Sn formula. |
| 3 | Find an using an = Sn - Sn-1 for n > 1. |
| 4 | Verify if the an formula from step 3 also holds true for n=1. |
By following these steps, you can effectively determine the general term of a sequence from its partial sums.
