The **sum** of the first **11** **terms** of the **arithmetic** **series** with the first term $a_1=-12$ and **common** **difference** $d=5$ can be determined using the **arithmetic** **series** sum formula. In an arithmetic sequence, each term after the first is found by adding the common difference to the preceding term.

This **characteristic** leads to a predictable pattern of numbers that have a constant difference between each pair of consecutive terms. To find the sum of the first 11 terms of such a series, I’ll use the **formula**: $S_n = \frac{n}{2}(2a_1 + (n-1)d)$, where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, $d$ is the **common** **difference**, and $n$ is the number of terms.

To add a bit of context, the concept of **arithmetic** **sequences** is fundamental in the study of **algebra** and is frequently encountered in various **mathematical** problems and real-life situations.

**Mathematics** often showcases elegant patterns, and **arithmetic** **sequences** embody this beautifully—a simple and predictable progression that unfolds with each term. Join me as we explore the intriguing world of **arithmetic sequences** and uncover the sum of their terms.

## Calculating the Sum of Arithmetic Series

When I’m faced with the **challenge** of finding the sum of the first few terms of an **arithmetic** **series**, I always rely on a specific formula that takes into account the first term, the common difference, and the number of terms I wish to add.

An **arithmetic** **series** is simply the addition of the terms in an **arithmetic** **sequence**, where each term increases by a steady amount, known as the **common** **difference** (*d*).

For example, if I want to calculate the sum of the first **11** **terms** of the **series** where the first term (*a₁*) is -12 and the **common** **difference** (*d*) is 5, I would use the following formula:

$$ S_n = \frac{n}{2} [2a_1 + (n-1)d] $$

Where:

*Sₙ*is the sum of the first*n*terms*a₁*is the first term*d*is the common difference between the terms*n*is the number of terms I want to sum up

To find the last term (*aₙ*), which I need for some **alternative** formulas, I use:

$$ a_n = a_1 + (n-1)d $$

Let’s put these formulas into practice:

Variable | Value |
---|---|

a₁ | -12 |

n | 11 |

d | 5 |

Now, I calculate the sum (*S₁₁*):

$$ S_{11} = \frac{11}{2} [2(-12) + (11-1)5] $$

$$ S_{11} = \frac{11}{2} [-24 + 50] $$

$$ S_{11} = \frac{11}{2} [26] $$

$$ S_{11} = \frac{286}{2} $$

$$ S_{11} = 143 $$

Therefore, the **sum** of the first 11 terms of this **arithmetic** **series** is 143.

I keep in mind that while an **arithmetic** **sequence** calculator could streamline the process, comprehending and manually applying the formula creates a solid foundation for solving **arithmetic** **sequence** problems with solutions. It’s rewarding to see how such a simple pattern of adding a fixed number repeatedly can lead to a quick solution for this type of series.

## Arithmetic Sequences Example

In this example, we’re looking at **arithmetic sequences**, a type of **sequence** in math where the difference between consecutive terms is constant. I’ll demonstrate how to calculate the sum of the first 11 terms with a given first term and **common** **difference**.

Here, the first term, $ a_1 $, is (-12), and the **common** **difference**, ( d ), is (5). I can use the following **arithmetic sequence formula** for the ( n )-th term:

$a_n = a_1 + (n-1)d$

To find the 11th term $a_{11}$:

$a_{11} = -12 + (11-1)\times 5$

$a_{11} = -12 + 10\times 5 $

$a_{11} = -12 + 50 $

$a_{11} = 38$

Here’s a quick table showing the **sequence** **development**:

Index (n) | $a_n $ |
---|---|

1 | (-12) |

2 | (-12 + 5 = -7) |

… | … |

11 | (-12 + 50 = 38) |

The **equation** to **evaluate** the sum of the first ( n ) terms in an **arithmetic** **sequence** is:

$S_n = \frac{n}{2}(a_1 + a_n)$

Now, let’s calculate the sum of the first 11 terms$S_{11} $:

$S_{11} = \frac{11}{2}(-12 + 38)$

$S_{11} = \frac{11}{2}(26)$

$S_{11} = \frac{286}{2}$

$S_{11} = 143$

The sum of the first 11 terms of this **arithmetic** **sequence** is (143). Each term is determined by its **index**, the function of the sequence, and follows a pattern with a **constant difference**. This demonstrates how **arithmetic** **sequences** work, whether the numbers are **positive** or **negative**.

## Conclusion

In our exploration of **arithmetic** **sequences**, I’ve demonstrated how to find the sum of the first 11 terms of a specific series. Starting with the initial term, *a1*, set at -12 and a **common** **difference**, *d*, of 5, I applied the formula for the sum of an **arithmetic** **series**:

$$ S_n = \frac{n}{2}(2a_1 + (n – 1)d) $$

Here, *n* represents the number of terms. Substituting our values into the formula:

$$ S_{11} = \frac{11}{2}(2(-12) + (11 – 1) \cdot 5) $$

Simplifying the **arithmetic** inside the parentheses gives us the sum we’ve been looking for:

$$ S_{11} = \frac{11}{2}(-24 + 50) $$ $$ S_{11} = \frac{11}{2} \cdot 26 $$ $$ S_{11} = 11 \cdot 13 $$ $$ S_{11} = 143 $$

Therefore, the sum of the first 11 terms of this **arithmetic series** is 143. By understanding the general formula, I was able to calculate this sum quickly and efficiently. This process can be used for any arithmetic series, provided you know the first term and the **common** **difference**.