What is the next number in the series?
0 1 5 14 30 __
50
I see a recursive formula:
term(n) = term(n-1) + (n-1)^2 , for n > 1 , term(1) = 0
term(1) = 0
term(2) = 0 + 1^2 = 1
term(3) = 1 + 2^2 = 5
term(4) = 5 + 3^2 = 14
term(5) = 14 + 4^2 = 30
term(6) = 30 + 5^2 = 55
To determine the pattern and find the next number in the series, we need to examine the differences between consecutive terms. Let's calculate the differences:
1 - 0 = 1
5 - 1 = 4
14 - 5 = 9
30 - 14 = 16
Looking at the differences, we notice that they are not constant. However, by adding 3 to each difference, we get:
1 + 3 = 4
4 + 3 = 7
9 + 3 = 12
16 + 3 = 19
Now, let's use these new differences to find the differences between the differences (also known as the second order differences):
4 - 1 = 3
7 - 4 = 3
12 - 9 = 3
19 - 16 = 3
We observe that the second order differences are constant. Since the second order differences are all equal to 3, we can conclude that the series is generated by a quadratic equation.
To find the next term, we can apply the quadratic equation formula. Let's assume the equation is of the form:
f(x) = ax² + bx + c
Using the differences, we can find the values of a, b, and c:
First order difference = 3
a + b + c = 3 (Equation 1)
Second order difference = 3
2a = 3 (Equation 2)
Solving Equation 2 for a gives us a = 3/2.
Substituting this value of a into Equation 1, we get:
(3/2) + b + c = 3
b + c = 3 - 3/2
b + c = 3/2
Now, we need additional information to find the values of b and c. If we consider the first term in the series (0), we can plug it into the quadratic equation:
0 = a(0)² + b(0) + c
0 = c
Therefore, c = 0.
Substituting c = 0 into b + c = 3/2, we obtain:
b + 0 = 3/2
b = 3/2
So, we have a = 3/2, b = 3/2, and c = 0.
Now, we can find the next number in the series by plugging in a value for x into the quadratic equation:
f(x) = (3/2)x² + (3/2)x
Let's find the value of f(6) to get the next term:
f(6) = (3/2)(6)² + (3/2)(6)
f(6) = (3/2)(36) + 9
f(6) = 54 + 9
f(6) = 63
Therefore, the next number in the series is 63.