Given the quadratic : 2;5; 10; 17
. Calculate the n^th term of the quadratic sequence
does your notation mean that
A2 = 5
A10 = 17
?? If so, two points do not determine a quadratic function.
Given the quadratic : 2;5; 10; 17
-- not familiar with that notation, are you saying you have the points
(1,2), (2,5), (3,10), (4,17) satisfying a quadratic function?
A quick observation will show that the y value is 1 more than the square of the x value,
that is .... (there are algorithms for more complicated values)
y = x^2 + 1
or
the nth term would be n^2 + 1
To calculate the nth term of a quadratic sequence, we first need to determine the formula for the sequence.
Given the quadratic sequence: 2, 5, 10, 17, ...
From the given sequence, we can observe that the differences between consecutive terms are: 3, 5, 7, ...
As the second differences are constant, we know that we have a quadratic sequence. A quadratic sequence can be written in the form:
an² + bn + c
To find the values of a, b, and c, we can substitute any three terms from the sequence into the equation and solve the resulting system of equations.
Let's use the three terms: (1) 2, (2) 5, (3) 10.
Substituting (1) into the equation:
a(1)² + b(1) + c = 2
a + b + c = 2
Substituting (2) into the equation:
a(2)² + b(2) + c = 5
4a + 2b + c = 5
Substituting (3) into the equation:
a(3)² + b(3) + c = 10
9a + 3b + c = 10
Now we have a system of three equations with three unknowns. We can use any method to solve this system, such as substitution or elimination. For simplicity, let's solve it by elimination.
Subtracting the first equation from the second equation:
(4a + 2b + c) - (a + b + c) = 5 - 2
3a + b = 3
Subtracting the first equation from the third equation:
(9a + 3b + c) - (a + b + c) = 10 - 2
8a + 2b = 8
Now we have a system of two equations with two unknowns, which is easier to solve. Solving this system, we find that a = 1 and b = 0.
Substituting these values back into the first equation:
1 + 0 + c = 2
c = 1
Therefore, the equation for the quadratic sequence is:
an² + bn + c = n² + 1
Now that we have the formula for the sequence, we can calculate the nth term by substituting the value of n into the equation.