In Pre-Calculus, it’s important to understand sequences and series. Two key types of sequences are arithmetic and geometric sequences. Let’s break down how to find terms in these sequences using simple formulas.

Arithmetic Sequence

An arithmetic sequence has a constant difference between each term.

• First Term: $a_1$
• Common Difference: $d$

To find the $n^{th}$ term ($a_n$) in an arithmetic sequence, you can use this formula:

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

Example:

Let's say the first term ($a_1$) is 3, and the common difference ($d$) is 2.

The first five terms of this sequence would be:

• $a_1 = 3$
• $a_2 = 5$
• $a_3 = 7$
• $a_4 = 9$
• $a_5 = 11$

Geometric Sequence

A geometric sequence has a constant ratio between each term.

• First Term: $g_1$
• Common Ratio: $r$

To find the $n^{th}$ term ($g_n$) in a geometric sequence, you can use this formula:

$$g_n = g_1 \cdot r^{(n - 1)}$$

Example:

If the first term ($g_1$) is 2, and the common ratio ($r$) is 3, then the first five terms will be:

• $g_1 = 2$
• $g_2 = 6$
• $g_3 = 18$
• $g_4 = 54$
• $g_5 = 162$

These formulas help you find and calculate the terms in both arithmetic and geometric sequences easily.