Given the quadratic : 2; 5; 10; 17 ..
. Calculate the n^th term of the quadratic sequence
https://owlcation.com/stem/Quadratic-Sequences-The-nth-term-of-a-quadratic-number-sequence
To find the nth term of a quadratic sequence, we need to determine the general formula for the sequence.
Looking at the given sequence: 2; 5; 10; 17, we notice that the difference between consecutive terms is increasing. The differences between consecutive terms are: 5 - 2 = 3, 10 - 5 = 5, and 17 - 10 = 7.
To find the general formula for a quadratic sequence, we need to examine the differences between consecutive terms. In a quadratic sequence, the differences are not constant, but they form a linear progression.
Since the differences between the terms form a linear progression, we take the second differences: 5 - 3 = 2, 7 - 5 = 2. We notice that the second differences are constant, which suggests a quadratic relationship.
Now, let's denote the sequence as an = f(n). From the given terms, we can write the following equations:
f(1) = 2
f(2) = 5
f(3) = 10
f(4) = 17
To find the general formula, we need to determine the quadratic equation in standard form: f(n) = an^2 + bn + c.
We can substitute the values of n and f(n) into the quadratic equation to get a system of equations:
a(1)^2 + b(1) + c = 2
a(2)^2 + b(2) + c = 5
a(3)^2 + b(3) + c = 10
a(4)^2 + b(4) + c = 17
Simplifying these equations, we get:
a + b + c = 2 --> Equation 1
4a + 2b + c = 5 --> Equation 2
9a + 3b + c = 10 --> Equation 3
16a + 4b + c = 17 --> Equation 4
Now we can solve this system of equations to find the values of a, b, and c.
By solving this system, we find that a = 2, b = -1, and c = 1.
Finally, the general formula for the sequence is:
f(n) = 2n^2 - n + 1
To find the nth term, you substitute the value of n into this formula.