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: Use two terms that are three steps apart.
Instead of comparing consecutive terms, use $a_1 = -\frac{5}{6}$ and $a_4 = -\frac{7}{12}$, since $a_4 - a_1 = 3d$.
Step 2: Solve for the common difference.
\[ 3d = -\frac{7}{12} - \left(-\frac{5}{6}\right) = -\frac{7}{12} + \frac{10}{12} = \frac{3}{12} = \frac{1}{4} \]
So $d = \frac{1}{12}$.
Step 3: Find the 31st term.
\[ a_{31} = a_1 + 30d = -\frac{5}{6} + \frac{30}{12} = -\frac{5}{6} + \frac{5}{2} = \frac{-5+15}{6} = \frac{10}{6} \]
\[ \boxed{\frac{5}{3}} \]