Determine the pattern: What comes next?
.1, 1, 2, 4, 7, _
.4, 9, 16, 25,_
Explain. I need help explaining. Thank you.
First looks like a sort of Fibonacci sequence, where each term is the sum of the three previous terms, making the sequence
1,1,2,4,7,13,24,44,...
However, that assumes a leading term of zero.
The leading .1 throws me off.
Or, it could be that each term is one less than a Fibonacci number:
Fib: 1 2 3 5 8 13 21 34 ...
You: 1 1 2 4 7 12 20 33 ...
The second one is obviously a sequence of perfect squares. Next is 36.
To determine the pattern and find what comes next in each sequence, we need to look for the relationship between the given numbers.
First sequence: .1, 1, 2, 4, 7
If we observe the differences between consecutive numbers, we get the following pattern:
1 - 0.1 = 0.9
2 - 1 = 1
4 - 2 = 2
7 - 4 = 3
We can see that the differences are increasing by 1 each time. So, to get the next difference, we add 1 to the current difference, which would be 3 + 1 = 4.
Now, to find the next number in the sequence, we add the next difference to the last number:
7 + 4 = 11
Therefore, the next number in the first sequence is 11.
Second sequence: .4, 9, 16, 25
If we observe the numbers themselves, we can see that they are perfect squares:
√.4 = 0.632
√9 = 3
√16 = 4
√25 = 5
We can determine that the next perfect square is 36. Therefore, the next number in the second sequence is 36.
To summarize:
- In the first sequence, the next number is 11.
- In the second sequence, the next number is 36.