How do you find the common difference with a sum?

To find the common difference (d) of an arithmetic sequence when you know the sum (S_n) of the first n terms, you need additional information, typically the first term (a₁) and the number of terms (n). You can then rearrange the formula for the sum of an arithmetic series to solve for d.

Understanding the Formula

The sum of the first n terms of an arithmetic sequence is given by:

S_n = n/2 [2a₁ + (n-1)d]

Where:

• S_n is the sum of the first n terms
• n is the number of terms
• a₁ is the first term
• d is the common difference

Steps to Find the Common Difference

Here's how to find d if you know S_n, n, and a₁:

1. Start with the formula: S_n = n/2 [2a₁ + (n-1)d]
2. Multiply both sides by 2/n: (2S_n)/n = 2a₁ + (n-1)d
3. Subtract 2a₁ from both sides: (2S_n)/n - 2a₁ = (n-1)d
4. Divide both sides by (n-1): d = [(2S_n)/n - 2a₁] / (n-1)
5. Simplify (optional): d = (2S_n - 2na₁) / [n(n-1)]

Example

Let's say you have an arithmetic series where:

• S_n = 40 (The sum of the first n terms is 40)
• n = 5 (There are 5 terms)
• a₁ = 2 (The first term is 2)

Now, let's find d:

d = [(2 40) / 5 - 2 2] / (5 - 1)
d = [80 / 5 - 4] / 4
d = [16 - 4] / 4
d = 12 / 4
d = 3

Therefore, the common difference is 3.

Alternative Forms & When to Use Them

If, instead of a₁, you know the last term (a_n), you can use the formula:

S_n = n/2(a₁ + a_n)

However, you would still need either a₁ or a_n, n, and S_n, along with the formula a_n = a₁ + (n-1)d to derive the common difference. This approach would involve solving a system of equations. The initial method is generally more direct if a₁ is already known.