Find the next term in the following number series :
2, 7, 24, 77, 238, .....

Question

Find the next term in the following number series : 
2, 7, 24, 77, 238, .....

Accepted Answer

To find the next term in the series 2, 7, 24, 77, 238, we need to identify the pattern or rule governing the sequence. Let's analyze the series step by step:

1. **Identify the pattern:**
   - The first term is 2.
   - The second term is 7.
   - The third term is 24.
   - The fourth term is 77.
   - The fifth term is 238.

2. **Look for a relationship between consecutive terms:**
   - From 2 to 7: $7 = 2 \times 3 + 1$
   - From 7 to 24: $24 = 7 \times 3 + 3$
   - From 24 to 77: $77 = 24 \times 3 + 5$
   - From 77 to 238: $238 = 77 \times 3 + 7$

Observing the pattern, each term is obtained by multiplying the previous term by 3 and then adding an odd number that increases by 2 each time (1, 3, 5, 7, ...).

3. **Apply the pattern to find the next term:**
   - The next odd number after 7 is 9.
   - Therefore, the next term should be calculated as follows:
     $$
     \text{Next term} = 238 \times 3 + 9 = 714 + 9 = 723
     $$

4. **Explanation of why Option 3 is correct:**
   - Based on the identified pattern, the next term in the series is 723, which matches Option 3.

5. **Explanation of why other options are incorrect:**
   - **Option 1 (721):** This does not fit the pattern, as 238 multiplied by 3 plus 9 equals 723, not 721.
   - **Option 2 (722):** Similarly, this does not fit the pattern, as the correct calculation results in 723.
   - **Option 4 (733):** This exceeds the calculated result of 723, thus not fitting the established pattern.

In conclusion, the correct answer is Option 3 (723), as it follows the logical pattern established in the series.