To find the first term of an arithmetic sequence, you should first understand its definition. An **arithmetic** **sequence** is a series of numbers in which the difference between consecutive terms is constant, known as the **common** **difference**.

If you’re given a term in the **sequence** other than the first, along with the **common** **difference**, you can calculate the first term by working backward using the general nth term formula: $a_n = a_1 + (n – 1)d$, where $a_n$ is the term you know, $a_1$ is the first term you’re solving for, $n$ is the position of the term in the **sequence**, and $d$ is a **common** **difference**.

I find that once you can identify these **components** in a problem, the rest becomes a simple matter of **substituting** values and solving for $a_1$. It’s like having a puzzle with all the pieces laid out in front of you; finding the first term is the key piece that helps you see the full picture. Are you ready to unlock the secrets of **arithmetic** **sequences**? Let’s do this together!

## Calculating the First Term In Arithmetic Sequence

An **arithmetic sequence** is a list of numbers with a specific pattern where each term is derived by adding a constant value, called the * common difference*, to the previous term. To identify the first term of an

**arithmetic**

**sequence**, we often use the

**nth term formula**:

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

Here, $a_n$ represents the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the **common difference**. When you have multiple terms of the sequence, the **common difference** ($d$) can be calculated by subtracting any term from the following term.

Assuming we know the second term ($a_2$) and the **common** **difference**, we could express the second term as:

$$a_2 = a_1 + d$$

Rearrange this to find the first term:

$$a_1 = a_2 – d$$

For example, if the second term is 7 and the common difference is 3, the first term is:

$$a_1 = 7 – 3$$ $$a_1 = 4$$

If the **common** **difference** is unknown but we have two non-consecutive terms, we can create a system of **equations** to solve for both $a_1$ and $d$. Let’s put this into a table:

Term Number (n) | Term Value ($a_n$) |
---|---|

1 | $a_1$ |

k | $a_k$ |

Using the formula for the $k^{th}$ term:

$$a_k = a_1 + (k-1)d$$

Through this, we can find the missing elements by solving the system of equations. In **practice**, this can be more complex and may require additional steps, which is why a **sequence calculator** can be a very useful tool. These calculators often require that you input the known terms of the sequence, and they will compute the missing values, including the first term.

## Conclusion

In this guide, I’ve outlined the steps to find the first term of an **arithmetic** **sequence**. To recap, when you have a specific term, say the $n^{th}$ term $a_n$, and the **common** **difference** (( d )), you can calculate the first term $a_1$ by rearranging the formula for finding the $n^{th}$ term:

$a_n = a_1 + (n – 1)d$

From this, the formula for the first term is:

$a_1 = a_n – (n – 1)d$

Using this equation ensures accuracy in pinpointing the **sequence’s** starting point. It’s important to have the value of one term and the **common** **difference** to efficiently determine the first term.

Remember that in **mathematics**, each step taken is critical for the integrity of your results. Precision and attention to detail can’t be overstated. I hope my guidance has made the process straightforward and less daunting for you.

If you’re ever in doubt, double-check your inputs and the **arithmetic** **operations** you perform. A small error can lead to significantly different outcomes. I encourage you to practice with different sequences as it is the best way to become confident in your ability to work with **arithmetic** **sequences**.