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: Understanding the Question:
We are given an Arithmetic Progression (A.P.) consisting of three terms:
\[ a_1 = -\frac{20}{9} \]
\[ a_2 = -\frac{2}{9} \]
\[ a_3 = \frac{16}{9} \]
We need to find the next term (which is the fourth term, \(a_4\)) of this sequence.

Step 2: Key Formula or Approach:
In an Arithmetic Progression, the difference between consecutive terms is constant. This is called the common difference, \(d\).
The common difference is given by:
\[ d = a_2 - a_1 = a_3 - a_2 \]
Once the common difference \(d\) is calculated, the next term \(a_4\) can be found using:
\[ a_4 = a_3 + d \]

Step 3: Detailed Explanation:

• Let us calculate the common difference \(d\) from the first two terms:
\[ d = a_2 - a_1 \]
\[ d = -\frac{2}{9} - \left(-\frac{20}{9}\right) \]
\[ d = -\frac{2}{9} + \frac{20}{9} \]
\[ d = \frac{-2 + 20}{9} = \frac{18}{9} = 2 \]

• Let us verify \(d\) with the next consecutive terms to ensure consistency:
\[ d = a_3 - a_2 \]
\[ d = \frac{16}{9} - \left(-\frac{2}{9}\right) \]
\[ d = \frac{16}{9} + \frac{2}{9} = \frac{18}{9} = 2 \]
- The common difference is verified as \(d = 2\) (or \(\frac{18}{9}\)).

• Now, calculate the fourth term \(a_4\) of the progression by adding \(d\) to the third term \(a_3\):
\[ a_4 = a_3 + d \]
\[ a_4 = \frac{16}{9} + 2 \]

• Convert 2 to a fraction with a denominator of 9:
\[ 2 = \frac{18}{9} \]
\[ a_4 = \frac{16}{9} + \frac{18}{9} \]
\[ a_4 = \frac{16 + 18}{9} = \frac{34}{9} \]


Step 4: Final Answer:
The next term of the given sequence is \(\frac{34}{9}\). This corresponds to option (D).