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, we need to analyze the pattern of differences between consecutive terms.
Let's start by calculating the differences between adjacent terms:
15 - 6 = 9
24 - 15 = 9
39 - 24 = 15
60 - 39 = 21
Now, let's calculate the differences between consecutive differences:
9 - 9 = 0 (since the first two differences were the same)
15 - 9 = 6
21 - 15 = 6
Since the second-row differences are now equal, it suggests that the sequence might be described by a quadratic equation.
Let's assume the formula for the sequence is:
f(n) = a + bn + cn(n-1)/2!
To find the values of a, b, and c, we can plug in the first three terms of the sequence into the formula. In this case, we'll use f(1), f(2), and f(3).
For the first term (n = 1):
6 = a + b(1) + c(1)(1-1)/2!
6 = a + b
For the second term (n = 2):
15 = a + b(2) + c(2)(2-1)/2!
15 = a + 2b + c
For the third term (n = 3):
24 = a + b(3) + c(3)(3-1)/2!
24 = a + 3b + 3c
Now we have a system of equations.
Equation 1: 6 = a + b
Equation 2: 15 = a + 2b + c
Equation 3: 24 = a + 3b + 3c
We can solve this system of equations to find the values of a, b, and c.
Subtract Equation 1 from Equation 2 to eliminate a:
15 - 6 = (a + 2b + c) - (a + b)
9 = b + c
Subtract Equation 2 from Equation 3 to eliminate a:
24 - 15 = (a + 3b + 3c) - (a + 2b + c)
9 = b + 2c
Now we have a new system of equations.
Equation 4: 9 = b + c
Equation 5: 9 = b + 2c
Subtract Equation 4 from Equation 5 to eliminate b:
9 - 9 = (b + 2c) - (b + c)
0 = c
Substitute the value of c = 0 into Equation 4 or Equation 5:
b + 0 = 9
b = 9
Now that we have found the values of b and c, we can substitute them back into any of the earlier equations to find a. Let's use Equation 1:
6 = a + 9
a = -3
Therefore, the formula for the sequence 6, 15, 24, 39, 60 is:
f(n) = -3 + 9n + 0(n(n-1))/2!
Simplifying the formula, we get:
f(n) = -3 + 9n
So, the formula for the given sequence is f(n) = -3 + 9n.