Question

The 31st term of the A.P. : \(-\frac{5}{6}, -\frac{3}{4}, -\frac{2}{3}, -\frac{7}{12}, \dots\) is

• A. \(-\frac{5}{3}\)
• B. \(\frac{5}{3}\)
• C. \(\frac{12}{20}\)
• D. \(-\frac{12}{20}\)

Hint:

Be careful with the signs when calculating the common difference \(d\).
Writing down the step-by-step LCM calculation helps avoid simple calculation mistakes.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given an Arithmetic Progression (A.P.) with fractional terms and we need to calculate its \(31\text{-st}\) term.

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

Step 3: Detailed Explanation:

• Identify the first term \(a\):
\[ a = -\frac{5}{6} \]

• Find the common difference \(d\) by subtracting the first term from the second term:
\[ d = \left(-\frac{3}{4}\right) - \left(-\frac{5}{6}\right) \]
\[ d = -\frac{3}{4} + \frac{5}{6} \]
To add these, find a common denominator (LCM of 4 and 6 is 12):
\[ d = \frac{-9 + 10}{12} = \frac{1}{12} \]

• Verify the difference with the next term:
\[ -\frac{2}{3} - \left(-\frac{3}{4}\right) = -\frac{8}{12} + \frac{9}{12} = \frac{1}{12} \]
The common difference is indeed \(d = \frac{1}{12}\).

• Apply the \(n\text{-th}\) term formula for \(n = 31\):
\[ a_{31} = a + 30d \]
\[ a_{31} = -\frac{5}{6} + 30 \left(\frac{1}{12}\right) \]

• Simplify the expression:
\[ a_{31} = -\frac{10}{12} + \frac{30}{12} \]
\[ a_{31} = \frac{20}{12} \]
Reduce the fraction by dividing the numerator and denominator by 4:
\[ a_{31} = \frac{5}{3} \]


Step 4: Final Answer:
The 31st term of the A.P. is \(\frac{5}{3}\).