Below is a sequence an, for n = 0, 1, 2, ...
n 0 1 2 3 4 5 6 7 8 9
an 25 25 19 25 61 145 295 529 865 1321
Use the method of finite differences to find a closed formula for this sequence.
To find a closed formula for this sequence using the method of finite differences, we need to examine the differences between consecutive terms in the sequence and their consecutive differences. Here's how we can do it step-by-step:
Step 1: Calculate the first differences:
Take the differences between consecutive terms in the sequence:
n 0 1 2 3 4 5 6 7 8 9
an 25 25 19 25 61 145 295 529 865 1321
1st difference: 0 -6 6 36 84 150 234 336 456
Probably hard to calculate in your head, so write them down
Step 2: Calculate the second differences:
Now, take the differences between consecutive terms of the 1st differences:
1st difference: 0 -6 6 36 84 150 234 336 456
2nd difference: -6 12 30 48 66 84 102 120
Step 3: Continue this process until we find a pattern or reach a constant difference:
1st difference: 0 -6 6 36 84 150 234 336 456
2nd difference: -6 12 30 48 66 84 102 120
3rd difference: 18 18 18 18 18 18 18 18
Step 4: Identify a pattern in the differences:
Looking at the differences, we notice that the 3rd difference is constant at 18.
Step 5: Formulate the closed formula:
Since the 3rd difference is constant, we can conclude that the original sequence can be described by a cubic polynomial. Let's assume the closed formula has the following form:
an = An^3 + Bn^2 + Cn + D
Step 6: Substitute values to find the coefficients:
Now, we can substitute the values of n and an from the original sequence to find the coefficients A, B, C, and D. We will choose four values to get a system of four equations.
For n = 0: 25 = D
For n = 1: 25 = A + B + C + D
For n = 2: 19 = 8A + 4B + 2C + D
For n = 3: 25 = 27A + 9B + 3C + D
Solving this system of equations, we find:
A = 1/3, B = 1/2, C = 5/6, D = 25
Step 7: Write the closed formula:
Now that we have found the values of the coefficients, we can express the closed formula for the sequence:
an = (1/3)n^3 + (1/2)n^2 + (5/6)n + 25
So, the closed formula for the given sequence is:
an = (1/3)n^3 + (1/2)n^2 + (5/6)n + 25