Question

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

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

Hint:

For multiple-choice questions on general terms, you can quickly test $n = 1$ in the options:
For Option (A): $3(1) - 4 = -1 \neq -\frac{1}{3}$.
For Option (B): $1 - \frac{4}{3} = -\frac{1}{3}$ (Matches first term).
For Option (C): $\frac{1 - 2}{3} = -\frac{1}{3}$ (Matches first term).
Now test $n = 2$ for Option (B) and (C):
For Option (B): $2 - \frac{4}{3} = \frac{2}{3}$ (Matches second term).
For Option (C): $\frac{2 - 2}{3} = 0 \neq \frac{2}{3}$.
This substitution method is extremely fast and reliable.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question is from the topic of "Arithmetic Progressions (A.P.)".
An Arithmetic Progression is a sequence of numbers in which the difference between consecutive terms is constant.
We are given a specific sequence of fractional terms and need to find the formula for its general or $n^{\text{th}}$ term.

Step 2: Key Formula or Approach:
The general term or $n^{\text{th}}$ term of an Arithmetic Progression is given by the formula:
\[ a_n = a + (n - 1)d \] where:

• $a$ is the first term of the A.P.

• $d$ is the common difference, calculated as $d = a_2 - a_1$.

• $n$ represents the position of the term in the sequence.

Step 3: Detailed Explanation:

• Identify the first term ($a$) of the given progression:
\[ a = -\frac{1}{3} \]

• Find the common difference ($d$) by subtracting the first term from the second term:
\[ d = \frac{2}{3} - \left(-\frac{1}{3}\right) = \frac{2}{3} + \frac{1}{3} = \frac{3}{3} = 1 \]

• Verify the common difference with subsequent terms to confirm it is constant:
\[ a_3 - a_2 = \frac{5}{3} - \frac{2}{3} = \frac{3}{3} = 1 \] \[ a_4 - a_3 = \frac{8}{3} - \frac{5}{3} = \frac{3}{3} = 1 \] Thus, the sequence is indeed an A.P. with $d = 1$.

• Substitute $a = -\frac{1}{3}$ and $d = 1$ into the general formula:
\[ a_n = -\frac{1}{3} + (n - 1)(1) \]

• Simplify the algebraic expression:
\[ a_n = n - 1 - \frac{1}{3} \] Combine the constant terms:
\[ -1 - \frac{1}{3} = -\frac{4}{3} \] \[ a_n = n - \frac{4}{3} \]

• Note that this expression can also be written with a common denominator as:
\[ a_n = \frac{3n - 4}{3} \] Both representations are mathematically identical.

Step 4: Final Answer:
The $n^{\text{th}}$ term is $n - \frac{4}{3}$ (or equivalently $\frac{3n - 4}{3}$), which corresponds to Option (B) and Option (D). Since both options are algebraically identical, choosing Option (B) is fully correct.