Question

\(n^{\text{th}}\) term of the A.P. : \(-\frac{1}{3}, \frac{4}{3}, 3, \ldots\) is

• A. \(\frac{5n - 9}{3}\)
• B. \(\frac{5n - 6}{3}\)
• C. \(\frac{3n - 4}{3}\)
• D. \(\frac{3n + 2}{3}\)

Hint:

An easy way to double check your answer is to substitute \(n = 1\) and \(n = 2\) into the final expression: - For \(n = 1\): \(\frac{5(1) - 6}{3} = -\frac{1}{3}\) (matches the first term).
- For \(n = 2\): \(\frac{5(2) - 6}{3} = \frac{4}{3}\) (matches the second term).
This confirms the result is correct!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given an Arithmetic Progression (A.P.): \[ -\frac{1}{3}, \frac{4}{3}, 3, \ldots \] We need to find the algebraic expression for the \(n^{\text{th}}\) term of this progression.

Step 2: Key Formula or Approach:
The general formula for the \(n^{\text{th}}\) term (\(a_n\)) of an Arithmetic Progression is: \[ a_n = a + (n - 1)d \] where: - \(a\) is the first term of the A.P.
- \(d\) is the common difference.
The common difference can be calculated as: \[ d = a_2 - a_1 \]

Step 3: Detailed Explanation:
1. Identify the first term \(a\) from the given series: \[ a = -\frac{1}{3} \] 2. Find the common difference \(d\) by subtracting the first term from the second term: \[ d = \frac{4}{3} - \left(-\frac{1}{3}\right) \] \[ d = \frac{4}{3} + \frac{1}{3} = \frac{5}{3} \] To verify, let us check the difference between the third term and the second term: \[ 3 - \frac{4}{3} = \frac{9}{3} - \frac{4}{3} = \frac{5}{3} \] The common difference is indeed \(d = \frac{5}{3}\).
3. Substitute \(a = -\frac{1}{3}\) and \(d = \frac{5}{3}\) into the \(n^{\text{th}}\) term formula: \[ a_n = -\frac{1}{3} + (n - 1)\frac{5}{3} \] \[ a_n = \frac{-1 + 5(n - 1)}{3} \] \[ a_n = \frac{-1 + 5n - 5}{3} \] \[ a_n = \frac{5n - 6}{3} \]

Step 4: Final Answer:
The \(n^{\text{th}}\) term of the given Arithmetic Progression is \(\frac{5n - 6}{3}\).
Thus, the correct option is (B).