Let a be the value of the function when n is 0. The number of the first row of differences is b, the first number of the next row is c, and so on. The formula for the nth number of the series is:
a + bn + cn(n-1)/2! + dn(n-l)(n-2)/3! and so on.
Find the formula for each pattern.
Here is the one I'm having trouble with:
6, 15, 24, 39, 60.
Please help if you can!
Thank you! :-)
To find the formula for the given sequence (6, 15, 24, 39, 60), we need to identify the pattern in the differences between terms.
Step 1: Find the first row of differences
The first row of differences is calculated by finding the difference between consecutive terms of the sequence:
15 - 6 = 9
24 - 15 = 9
39 - 24 = 15
60 - 39 = 21
The first row of differences is: 9, 9, 15, 21
Step 2: Find the second row of differences
Similarly, we find the differences between consecutive terms of the first row of differences:
9 - 9 = 0
15 - 9 = 6
21 - 15 = 6
The second row of differences is: 0, 6, 6
Step 3: Find the third row of differences (optional)
If necessary, continue calculating the differences until you reach a row where all the numbers are the same. In this case, it is not necessary, as the third row of differences is not needed to determine the formula.
Step 4: Identify the pattern in the differences
From the first row of differences, we can observe that there is no constant difference between terms. However, from the second row of differences, we can see that there is a constant difference of 6 between terms.
Step 5: Find the formula
To determine the formula for the sequence, we plug in the values of a = 6, b = 9 (the first number of the first row of differences), and c = 6 (the constant difference from the second row of differences) into the provided formula:
n(n+1)/2
The formula for the given sequence is:
6 + 9n + 6n(n+1)/2
Simplifying, we get:
6 + 9n + 3n(n+1)
which further simplifies to:
6 + 9n + 3n^2 + 3n
Combining like terms, we get:
3n^2 + 12n + 6
So, the formula for the sequence (6, 15, 24, 39, 60) is:
3n^2 + 12n + 6