What is the sum sn of the first n terms of an arithmetic sequence is given by the formula sn?

The sum S_n of the first n terms of an arithmetic sequence is given by the formula: S_n = n/2 [2a₁ + (n - 1)d] or S_n = n/2 (a₁ + a_n), where a₁ is the first term, a_n is the nth term, and d is the common difference.

Understanding the Formulas

There are two common formulas to calculate the sum of an arithmetic series. Let's break them down:

• *Formula 1: S_n = n/2 [2a₁ + (n - 1)d]**

  • S_n represents the sum of the first n terms.
  • n is the number of terms you are summing.
  • a₁ is the first term of the arithmetic sequence.
  • d is the common difference between consecutive terms.
  • This formula is particularly useful when you know the first term (a₁) and the common difference (d).

• *Formula 2: S_n = n/2 (a₁ + a_n)**

  • S_n represents the sum of the first n terms.
  • n is the number of terms you are summing.
  • a₁ is the first term of the arithmetic sequence.
  • a_n is the nth term of the arithmetic sequence.
  • This formula is beneficial when you know the first term (a₁) and the last term (a_n) you want to sum. The provided reference states this formula as Sn=2n(a1+an), but this is incorrect. The correct formula is Sn = n/2(a1 + an).

Example

Let's say we have an arithmetic sequence: 2, 4, 6, 8, 10.

We want to find the sum of the first 5 terms (n = 5).

• a₁ = 2 (the first term)
• a₅ = 10 (the fifth term)
• d = 2 (the common difference)

Using the second formula:

S₅ = 5/2 (2 + 10) = 5/2 12 = 30

Using the first formula:

S₅ = 5/2 [22 + (5-1)2] = 5/2 [4 + 8] = 5/2 * 12 = 30

Therefore, the sum of the first 5 terms is 30.

In summary, the sum of the first n terms of an arithmetic sequence is calculated using either S_n = n/2 [2a₁ + (n - 1)d] or S_n = n/2 (a₁ + a_n), depending on the given information.