Describe the pattern each sequence manifests.
-1,-3,-5,...___________
2,1,0,...________________
-1,2,3,...___________
-5,-2,1,4,...__________
-1,2,7,...________
3,8,15,..._________
keep adding -2 each time. a_n = -(2n-1)
The next three are just the same idea.
The last one is a bit different. Note that each term is one less than a power of 2.
- The first sequence is decreasing by 2 each time, so the pattern is subtract 2 from the previous number: -1, -3, -5, -7, ...
- The second sequence is decreasing by 1 each time, so the pattern is subtract 1 from the previous number: 2, 1, 0, -1, ...
- The third sequence is increasing by 1 each time, so the pattern is add 1 to the previous number: -1, 2, 3, 4, ...
- The fourth sequence is increasing by 3 each time, so the pattern is add 3 to the previous number: -5, -2, 1, 4, ...
- The fifth sequence is increasing by 5 each time, so the pattern is add 5 to the previous number: -1, 2, 7, 12, ...
- The sixth sequence is increasing by consecutive odd numbers: 3, 5, 7, 9, ... So the pattern is add consecutive odd numbers starting from 1 to the previous number: 3, 8, 15, 24, ...
To describe the pattern in each sequence, we need to analyze the difference between consecutive terms. Let's examine each sequence individually:
1. -1, -3, -5, ...
The pattern in this sequence is that each term is decreasing by 2. So, to get the next term, we subtract 2 from the previous term. The next term would be -7.
2. 2, 1, 0, ...
The pattern in this sequence is that each term is decreasing by 1. So, to get the next term, we subtract 1 from the previous term. The next term would be -1.
3. -1, 2, 3, ...
The pattern in this sequence is that the terms alternate between increasing by 1 and increasing by 2. So, to get the next term, we add 1 if the previous term was even, and add 2 if the previous term was odd. The next term would be 5.
4. -5, -2, 1, 4, ...
The pattern in this sequence is that each term is increasing by 3. So, to get the next term, we add 3 to the previous term. The next term would be 7.
5. -1, 2, 7, ...
The pattern in this sequence is that each term is increasing by the square of its position (1^2, 2^2, 3^2, ...). So, to get the next term, we add the square of the next position number to the previous term. The next term would be 14.
6. 3, 8, 15, ...
The pattern in this sequence is that each term is increasing by the sequence of odd numbers (2n-1). So, to get the next term, we add the next odd number (2n+1) to the previous term. The next term would be 24.
By examining the differences between consecutive terms and applying the appropriate arithmetic operations, we can identify the pattern in each sequence and determine the next term.