Sn N 2 2a1 N-1 D Calculator

The Sn = n/2 [2a1 + (n-1)d] formula calculates the sum of an arithmetic series. This calculator provides an easy way to compute the sum of a series given the first term, common difference, and number of terms.

What is the Sn = n/2 [2a1 + (n-1)d] formula?

The formula Sn = n/2 [2a1 + (n-1)d] is used to find the sum of the first n terms of an arithmetic series. An arithmetic series is a sequence of numbers where the difference between consecutive terms is constant.

Formula: Sn = n/2 [2a1 + (n-1)d]

Sn = Sum of the first n terms
n = Number of terms
a1 = First term of the series
d = Common difference between terms

This formula is derived from the fact that the sum of an arithmetic series can be visualized as the area of a trapezoid where the two parallel sides are the first and last terms of the series, and the height is the number of terms.

Note: This formula works only for arithmetic series where the difference between consecutive terms is constant. For geometric series, a different formula is used.

How to use this calculator

Enter the first term (a1) of your arithmetic series
Enter the common difference (d) between terms
Enter the number of terms (n) you want to sum
Click the "Calculate" button
View the result and chart visualization

The calculator will display the sum of the series (Sn) and show a chart of the series terms. You can also reset the form to start over.

Worked examples

Example 1: Simple arithmetic series

Given an arithmetic series with first term a1 = 2, common difference d = 3, and number of terms n = 5:

Term	Value
a1	2
a2	5
a3	8
a4	11
a5	14
Sum (Sn)	40

Using the formula: Sn = 5/2 [2*2 + (5-1)*3] = 5/2 [4 + 12] = 5/2 * 16 = 40

Example 2: Negative common difference

Given an arithmetic series with first term a1 = 10, common difference d = -2, and number of terms n = 4:

Term	Value
a1	10
a2	8
a3	6
a4	4
Sum (Sn)	28

Using the formula: Sn = 4/2 [2*10 + (4-1)*-2] = 2 [20 - 6] = 2 * 14 = 28

Frequently Asked Questions

What is an arithmetic series?: An arithmetic series is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference.
When should I use this formula?: Use this formula when you need to find the sum of the first n terms of an arithmetic series. It's particularly useful in mathematics, physics, and engineering problems involving sequences.
What if the common difference is zero?: If the common difference (d) is zero, all terms in the series are equal to the first term (a1). The sum will simply be n multiplied by a1.
Can I use this formula for infinite series?: No, this formula is specifically for finite arithmetic series. For infinite series, different convergence criteria must be considered.