How do you find the sum of the following arithmetic sequence?

To find the sum of an arithmetic sequence, you can use a specific formula that leverages the properties of arithmetic progressions.

Here's how to do it:

Understanding Arithmetic Sequences

An arithmetic sequence is a series of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference.

Formula for the Sum of an Arithmetic Sequence

The sum (S) of an arithmetic sequence can be calculated using the following formula:

S = (n/2) * (a₁ + a_n)

Where:

• S = Sum of the arithmetic sequence
• n = Number of terms in the sequence
• a₁ = First term of the sequence
• a_n = Last term of the sequence

Steps to Calculate the Sum

1. Identify a₁ and a_n: Determine the first term (a₁) and the last term (a_n) of the arithmetic sequence.
2. Determine n: Count the number of terms (n) in the sequence.
3. Apply the formula: Substitute the values of a₁, a_n, and n into the formula: S = (n/2) * (a₁ + a_n).
4. Calculate S: Perform the calculation to find the sum (S) of the arithmetic sequence.

Example

Let's say we want to find the sum of the arithmetic sequence: 2, 4, 6, 8, 10.

1. a₁ = 2 (First term)
2. a_n = 10 (Last term)
3. n = 5 (Number of terms)

Applying the formula:

S = (5/2) (2 + 10) = (5/2) 12 = 30

Therefore, the sum of the arithmetic sequence 2, 4, 6, 8, 10 is 30.

Alternative Method When the Last Term Is Unknown

If you don't know the last term (a_n) but you know the common difference (d), you can use another formula to find the sum:

S = (n/2) * [2a₁ + (n - 1)d]

Where:

• S = Sum of the arithmetic sequence
• n = Number of terms in the sequence
• a₁ = First term of the sequence
• d = Common difference

Summary

Finding the sum of an arithmetic sequence involves identifying the first term, last term, and the number of terms. Then, you can use the appropriate formula to calculate the sum efficiently.