Given the progression 3, 5, 7, 9,.... . . . find an expression for the (n - 2)$^{th}$ term of the progression.

1. Analyze the question

We are given an arithmetic progression (AP): 3, 5, 7, 9, ... and we need to find an expression for the $(n - 2)^{th}$ term.

2. Explain the concept

In an arithmetic progression, the $n^{th}$ term ($T_n$) is calculated using the formula:

Where:

• $a$ is the first term.
• $d$ is the common difference.
• $n$ is the position of the term.

For this sequence:

• $a = 3$
• $d = 5 - 3 = 2$

3. Show the solution

First, let's find the general formula for the $n^{th}$ term of this specific sequence:

Now, the question asks for the $(n - 2)^{th}$ term. To find this, we substitute $(n - 2)$ into the position variable of our general formula:

Expand the brackets:

Simplify the expression:

4. Identify the answer

Comparing our result ($2n - 3$) with the given options, it matches option D.

5. Exam Tip

When asked for a term like $(n - k)^{th}$, don't panic! Simply find the general $n^{th}$ term formula first, then replace every '$n$' in that formula with the expression $(n - k)$. Always remember to use brackets when substituting to avoid sign errors during expansion.