Question

The n\(^{th}\) term of the A.P. \(-\frac{1}{3}\), \(\frac{2}{3}\), \(\frac{5}{3}\), \(\frac{8}{3}\), ... is :

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

Hint:

An easy way to verify your answer is to substitute \( n = 1 \) and \( n = 2 \) into the options:
- For option (B), substituting \( n = 1 \) gives \( 1 - \frac{4}{3} = -\frac{1}{3} \), which matches the first term.
- Substituting \( n = 2 \) gives \( 2 - \frac{4}{3} = \frac{2}{3} \), which matches the second term.
This substitution test can be completed very quickly and confirms the correct answer without any risk of algebraic errors.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The topic of this question is Arithmetic Progressions (AP).
We are given an infinite arithmetic progression: \( -\frac{1}{3}, \frac{2}{3}, \frac{5}{3}, \frac{8}{3}, \dots \)
We need to find the formula for the general term (the \( n \)-th term, represented as \( a_n \)) of this progression.

Step 2: Key Formula or Approach:
The formula for the \( n \)-th term of an Arithmetic Progression is:
\[ a_n = a + (n - 1)d \]
Where:
- \( a \) is the first term of the AP.
- \( d \) is the common difference between consecutive terms.
- \( n \) is the position of the term in the progression.

Step 3: Detailed Explanation:
1. Identify the first term \( a \) of the given AP:
\[ a = -\frac{1}{3} \]
2. Calculate the common difference \( d \) by subtracting the first term from the second term:
\[ d = \frac{2}{3} - \left(-\frac{1}{3}\right) \]
Simplify the subtraction of fractions:
\[ d = \frac{2}{3} + \frac{1}{3} = \frac{2 + 1}{3} = \frac{3}{3} = 1 \]
3. Substitute the values of \( a = -\frac{1}{3} \) and \( d = 1 \) into the \( n \)-th term formula:
\[ a_n = -\frac{1}{3} + (n - 1) \times 1 \]
4. Simplify the expression algebraically:
\[ a_n = -\frac{1}{3} + n - 1 \]
5. Combine the constant numerical terms:
\[ a_n = n - 1 - \frac{1}{3} \]
Find a common denominator to subtract 1 and \(\frac{1}{3}\):
\[ -1 - \frac{1}{3} = -\frac{3}{3} - \frac{1}{3} = -\frac{4}{3} \]
6. Substitute this combined constant back into our expression:
\[ a_n = n - \frac{4}{3} \]
7. This matches option (B).

Step 4: Final Answer:
The \( n \)-th term of the AP is \(n - \frac{4}{3}\), which corresponds to option (B).