Understanding how to find the nth term in arithmetic and geometric sequences is really important for Year 9 Math. Let’s make this simple!

Arithmetic Sequences

An arithmetic sequence is a list of numbers where the difference between each number and the next is always the same. This difference is called the "common difference," and we often use the letter $d$ to represent it.

How to Find the nth Term:

To find the nth term ($T_n$) in an arithmetic sequence, you can use this formula:

$T_n = a + (n - 1)d$

Here’s what the letters mean:

• $T_n$ is the nth term you want to find.
• $a$ is the first term in the sequence.
• $d$ is the common difference.
• $n$ is the number of the term you're looking for.

Example:

Let’s look at the arithmetic sequence 2, 5, 8, 11, ...

In this sequence:

• The first term ($a$) is 2.
• The common difference ($d$) is 3 (because 5 - 2 = 3).

If we want to find the 10th term:

$T_{10} = a + (10 - 1)d = 2 + 9 \times 3 = 2 + 27 = 29$

So, the 10th term is 29.

Geometric Sequences

Now, a geometric sequence is a different kind of number list. In this sequence, you get each term by multiplying or dividing the previous term by a specific number called the "common ratio," which we call $r$.

How to Find the nth Term:

To find the nth term ($G_n$) in a geometric sequence, you can use this formula:

$G_n = a \cdot r^{(n - 1)}$

Let’s explain what the letters mean:

• $G_n$ is the nth term you want to find.
• $a$ is the first term in the sequence.
• $r$ is the common ratio.
• $n$ is the number of the term you're looking for.

Example:

Let's take the geometric sequence 3, 6, 12, 24, ...

In this sequence:

• The first term ($a$) is 3.
• The common ratio ($r$) is 2 (because 6 ÷ 3 = 2).

If we want to find the 5th term:

$G_5 = a \cdot r^{(5 - 1)} = 3 \cdot 2^4 = 3 \cdot 16 = 48$

So, the 5th term is 48.

Summary

To sum it up, finding the nth term in arithmetic and geometric sequences is all about using the right formulas.

For arithmetic sequences, remember:

• Formula: $T_n = a + (n - 1)d$

For geometric sequences, remember:

• Formula: $G_n = a \cdot r^{(n - 1)}$

With these formulas, you can easily calculate any term in these sequences! Just make sure to identify the first term and the common difference or ratio before using the formulas. Happy learning!