Calculating the Sum of an Arithmetic Series with a Given Middle Term

In an arithmetic series, the middle term is a critical value that can be used to find the sum of the first n terms. This article explores the process of calculating the sum of the first eleven terms of an arithmetic series given that the middle term is 30. We will apply the formulas and step-by-step reasoning to find the solution.

Understanding the Arithmetic Series and Middle Term

For an arithmetic series with 11 terms, the middle term (which is the 6th term) can be considered as the average of the first and last terms. Given that the middle term is 30, we denote the first term as a and the common difference as d. Therefore, the 6th term can be expressed as:

a 5d 30

Calculating the Sum of the First Eleven Terms

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

Sn

For our case, n 11:

S11

We can rewrite 2a 10d as follows:

2a 10d 2a 5d 2 * 30 60

Substituting this back into the sum formula:

S11

Therefore, the sum of the first eleven terms of the arithmetic series is 330.

Alternative Approach

The sum of the first 11 terms of a series that contains 11 terms is the same as the sum of the series itself. The middle term of an arithmetic series is the average arithmetic mean value of the series. The sum of the series is this average times the number of terms. In this case, the sum of the first 11 terms of the series is also the average times the number of terms. If it is an arithmetic series, the following holds:

∑i111 ai overline{a} * 11 30 * 11 330

Term Representation and Summing

We can represent the terms of the series as follows, starting from the middle term:

a1 a_6 - 5d, a2 a_6 - 4d, a3 a_6 - 3d, a4 a_6 - 2d, a5 a_6 - d, a6 a_6 - 0d, a7 a_6 d, a8 a_6 2d, a9 a_6 3d, a10 a_6 4d, a11 a_6 5d

Summing all the terms:

a1 a_2 a_3 a_4 a_5 a_6 a_7 a_8 a_9 a_10 a_11 (11a_6)

11 * 30 330

As we can see, the sum of all the terms exactly cancels out, leaving us with the product of the number of terms and the middle term. Therefore, the sum of the first eleven terms of the series is 330.