How to Find the First Term of an Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference, denoted by $d$. To find the first term of an arithmetic sequence, you can use the formula for the $n$-th term of an arithmetic sequence:

$a_n = a_1 + (n – 1) times d$

Here, $a_n$ is the $n$-th term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

Step-by-Step Process

1. Identify the Known Values

   • Determine the value of $a_n$ (the $n$-th term).
   • Determine the common difference $d$
   • Determine the term number $n$

2. Rearrange the Formula

   • To find the first term $a_1$, rearrange the formula:

   $a_1 = a_n – (n – 1) times d$

3. Substitute the Known Values

   • Plug in the values of $a_n$, $d$, and $n$ into the rearranged formula.

4. Solve for $a_1$

   • Perform the arithmetic operations to find the value of $a_1$

Example

Let’s say we have an arithmetic sequence where the 5th term ($a_5$) is 20, and the common difference ($d$) is 3. To find the first term ($a_1$), follow these steps:

1. Identify the known values:

   • $a_5 = 20$
   • $d = 3$
   • $n = 5$

2. Rearrange the formula:

   $a_1 = a_5 – (5 – 1) times 3$

3. Substitute the known values:

   $a_1 = 20 – 4 times 3$

4. Solve for $a_1$:

   $a_1 = 20 – 12 = 8$

So, the first term $a_1$ is 8.

Conclusion

Finding the first term of an arithmetic sequence is straightforward once you know the $n$-th term, the common difference, and the term number. By rearranging the formula for the $n$-th term and plugging in the known values, you can easily solve for the first term. This method is essential for understanding arithmetic sequences and their properties.