Determine the pattern: What comes next?
.1, 1, 2, 4, 7, _
.4, 9, 16, 25,_
Explain. I need help explaining. Thank you.
To determine the pattern for each sequence and figure out what comes next, we can analyze the differences between the terms.
Sequence 1: .1, 1, 2, 4, 7
By examining the differences between consecutive terms, we can see that:
1 - .1 = 0.9
2 - 1 = 1
4 - 2 = 2
7 - 4 = 3
The differences between the terms are not constant; they are increasing by 0.9, 1, 2, and 3 respectively. To determine the next difference, we can continue this pattern: 4 + 0.9 = 4.9.
To find the next term, we add the next difference of 4.9 to the last term in the sequence:
7 + 4.9 = 11.9
Therefore, the next term in the sequence is 11.9.
Sequence 2: .4, 9, 16, 25
Similarly, we can calculate the differences between consecutive terms:
9 - .4 = 8.6
16 - 9 = 7
25 - 16 = 9
In contrast to Sequence 1, the differences between the terms are not increasing or decreasing in a consistent manner. However, if we observe the resulting differences, they are consecutive odd numbers (8.6, 7, 9).
To find the next difference, we can continue this pattern by adding the next odd number, which is 11, to the previous difference of 9:
9 + 11 = 20
Lastly, we add the new difference of 20 to the last term in the sequence:
25 + 20 = 45
Therefore, the next term in the sequence is 45.
By analyzing the patterns and differences in the given sequences, we can determine that the next terms are 11.9 for Sequence 1 and 45 for Sequence 2.