Question: For a sequence if $S_n = \frac{n}{n+1}$. Find the sequence.
Solution:
To find the individual terms of a sequence when you are given the formula for the sum of its first $n$ terms ($S_n$), you can use the following relationship:
The $n$-th term ($a_n$) is the difference between the sum of the first $n$ terms and the sum of the first $n-1$ terms.
$$a_n = S_n - S_{n-1} \quad \text{for } n \ge 2$$
For the very first term, the sum of the first $1$ term is simply the first term itself:
$$a_1 = S_1$$
Step 1: Find the first term ($a_1$)
Substitute $n = 1$ into your given formula $S_n = \frac{n}{n+1}$:
$$a_1 = S_1 = \frac{1}{1 + 1} = \frac{1}{2}$$
Step 2: Find the formula for the $n$-th term ($a_n$)
First, determine the expression for $S_{n-1}$ by replacing $n$ with $n-1$ in the original formula:
$$S_{n-1} = \frac{n - 1}{(n - 1) + 1} = \frac{n - 1}{n}$$
Now, subtract $S_{n-1}$ from $S_n$:
$$a_n = \frac{n}{n+1} - \frac{n-1}{n}$$
To subtract these fractions, find the common denominator, which is $n(n+1)$:
$$a_n = \frac{n \cdot n}{n(n+1)} - \frac{(n-1)(n+1)}{n(n+1)}$$
$$a_n = \frac{n^2 - (n^2 - 1)}{n(n+1)}$$
$$a_n = \frac{n^2 - n^2 + 1}{n(n+1)}$$
$$a_n = \frac{1}{n(n+1)}$$
Step 3: Determine the sequence
By substituting $n = 1, 2, 3, 4, \dots$ into the general formula $a_n = \frac{1}{n(n+1)}$, the sequence is:
$$\frac{1}{2}, \frac{1}{6}, \frac{1}{12}, \frac{1}{20}, \dots$$