Calculate Sum with N A

An arithmetic series is a sequence of numbers where the difference between consecutive terms is constant. This calculator helps you find the sum of an arithmetic series when you know the first term (a), the common difference (d), and the number of terms (n).

What is an arithmetic series?

An arithmetic series is the sum of the terms in an arithmetic sequence. An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference (d).

For example, the sequence 2, 5, 8, 11, 14 is an arithmetic sequence where the first term (a) is 2 and the common difference (d) is 3. The sum of the first 5 terms of this sequence is 2 + 5 + 8 + 11 + 14 = 40.

Arithmetic series formula

The sum of the first n terms of an arithmetic series can be calculated using the following formula:

Sₙ = n/2 × (2a + (n - 1)d)

Where:

Sₙ is the sum of the first n terms
n is the number of terms
a is the first term
d is the common difference between terms

This formula works for any arithmetic series, whether the common difference is positive or negative.

How to calculate the sum of an arithmetic series

To calculate the sum of an arithmetic series using the formula, follow these steps:

Identify the first term (a), common difference (d), and number of terms (n).
Plug these values into the formula: Sₙ = n/2 × (2a + (n - 1)d).
Calculate the expression inside the parentheses: (2a + (n - 1)d).
Multiply the result by n/2 to get the sum Sₙ.

Note: The formula only works for arithmetic series. If your sequence is not arithmetic, you cannot use this formula.

Example calculation

Let's calculate the sum of the first 10 terms of an arithmetic series where the first term is 3 and the common difference is 2.

Using the formula:

S₁₀ = 10/2 × (2×3 + (10 - 1)×2)

S₁₀ = 5 × (6 + 18)

S₁₀ = 5 × 24

S₁₀ = 120

So, the sum of the first 10 terms is 120.

FAQ

What is the difference between an arithmetic sequence and an arithmetic series?: An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. An arithmetic series is the sum of the terms in an arithmetic sequence.
Can I use this formula for any arithmetic series?: Yes, the formula works for any arithmetic series, regardless of whether the common difference is positive or negative.
What if I don't know the number of terms?: If you know the first term, common difference, and the last term, you can calculate the number of terms using the formula for the nth term of an arithmetic sequence: aₙ = a + (n - 1)d. Then you can use the sum formula.
Is there a calculator that can help me with this?: Yes, this page includes an interactive calculator that can compute the sum of an arithmetic series for you.