Question

If \(\frac{-20}{9}, \frac{-2}{9}, \frac{16}{9}, \ldots\) are in A.P., then next term of the sequence is

• A. \(\frac{32}{9}\)
• B. \(\frac{46}{9}\)
• C. \(\frac{2}{9}\)
• D. \(\frac{34}{9}\)

Hint:

Notice the pattern in the numerators while keeping the common denominator 9:
The numerators are: \(-20, -2, 16, \ldots\)
The difference between successive numerators is:
\[ -2 - (-20) = 18 \]
\[ 16 - (-2) = 18 \]
So the next numerator must be:
\[ 16 + 18 = 34 \]
Therefore, the next term is \(\frac{34}{9}\). Working only with numerators is faster!

Answer 1

The Correct Option is D

Step 1: Identify the Given Terms.
The AP is $-\dfrac{20}{9},\ -\dfrac{2}{9},\ \dfrac{16}{9},\ \ldots$ We have $a_1 = -\dfrac{20}{9}$, $a_2 = -\dfrac{2}{9}$, $a_3 = \dfrac{16}{9}$.
Step 2: Find the Common Difference.
\[ d = a_2 - a_1 = -\frac{2}{9} - \left(-\frac{20}{9}\right) = -\frac{2}{9} + \frac{20}{9} = \frac{18}{9} = 2 \]
Step 3: Verify the Common Difference.
$a_3 - a_2 = \dfrac{16}{9} - \left(-\dfrac{2}{9}\right) = \dfrac{18}{9} = 2$. Great, the common difference is consistently $d = 2$.
Step 4: Find the Next (Fourth) Term.
\[ a_4 = a_3 + d = \frac{16}{9} + 2 = \frac{16}{9} + \frac{18}{9} = \frac{34}{9} \]
Step 5: Confirm the Pattern.
The sequence becomes $-\dfrac{20}{9},\ -\dfrac{2}{9},\ \dfrac{16}{9},\ \dfrac{34}{9}$. Each successive term increases by 2, so this is consistent.
Step 6: Match with Options.
$\dfrac{34}{9}$ corresponds to option (4).
\[ \boxed{\dfrac{34}{9}} \]