What is the next term in the sequence below

What is the next term in the sequence below?

0.25, 0.75, 2.25, 6.75, …
A. 6.25
B. 10.25
C. 20.25
D. 60.75

The Correct Answer and Explanation is :

To find the next term in the sequence (0.25, 0.75, 2.25, 6.75, \ldots), let’s first analyze the pattern in the given numbers.

1. Identifying the Differences:

• The first term is (0.25).
• The second term is (0.75).
• The difference between the second and first terms is (0.75 – 0.25 = 0.50).
• The third term is (2.25).
• The difference between the third and second terms is (2.25 – 0.75 = 1.50).
• The fourth term is (6.75).
• The difference between the fourth and third terms is (6.75 – 2.25 = 4.50).

Thus, we can summarize the differences:

• (0.50, 1.50, 4.50)

1. Finding the Pattern in the Differences:
   Next, let’s analyze the differences between these differences:

• The difference between (1.50) and (0.50) is (1.00).
• The difference between (4.50) and (1.50) is (3.00).

So we can summarize these differences as follows:

• The differences of differences: (1.00, 3.00)

1. Further Analysis:
   The second differences themselves do not form a simple arithmetic or geometric sequence. However, the sequence of terms appears to grow exponentially.
2. Finding the Next Term:
   If we observe the relationship of the terms in terms of multiplication:

• (0.25 \times 3 = 0.75)
• (0.75 \times 3 = 2.25)
• (2.25 \times 3 = 6.75)

It appears that each term is multiplied by (3) to obtain the next term. If we apply this pattern to find the next term:

• The next term would be (6.75 \times 3 = 20.25).

1. Conclusion:
   Thus, the next term in the sequence is C. 20.25. This shows that the sequence is created by multiplying each term by (3), confirming the exponential growth.