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

Answer Details

This sequence is an example of an arithmetic progression (AP), where each term increases by the same amount. In this case, each term increases by 2. Let's understand how to find an expression for any term in such a sequence, and specifically for the \((n-2)^{\text{th}}\) term.

Step 1: Identify the formula for the \(k^{\text{th}}\) term of an arithmetic progression
For an AP with a first term \(a\) and common difference \(d\), the general term (also called the nth term) is: \[ a_k = a + (k-1)d \]

Step 2: Find the values for this sequence
Given sequence: 3, 5, 7, 9, ...
Here, the first term (\(a\)) is 3 and the common difference (\(d\)) is 2 (because \(5-3 = 2\), \(7-5 = 2\), etc).

Step 3: Substitute into the formula
The general formula becomes: \[ a_k = 3 + (k-1)\times 2 = 3 + 2k - 2 = 2k + 1 \]

Step 4: Apply to the \((n-2)^{\text{th}}\) term
We are asked for the value at the \((n-2)^{\text{th}}\) place: \[ a_{n-2} = 2(n-2) + 1 \] Let's simplify this: \[ a_{n-2} = 2n - 4 + 1 = 2n - 3 \]

Why is this correct?
This formula represents the value in the sequence at position \((n-2)\). The \(n\) in the formula corresponds to the variable position you want to analyze. The arithmetic progression pattern and algebra simplify directly to \(2n - 3\), matching the structure of the sequence.

Summary Table for Clarity:

The expression for the \((n-2)^{\text{th}}\) term of this progression is therefore \(2n - 3\).