the sequence 4;9;x;37;...is a quadratic sequence. a) calculate x. b) determine nth term of the sequence
4, 9, x, 37
first difference:
5, x-9, 37-x
2nd difference
x-14, 37-x - (x-9)
x-14, 46-2x
To be a quadratic sequence, the 2nd differences must all be the same, so
x-14 = 46-2x
3x = 60
x = 20
so our terms are 4, 9, 20, 37
To find x in the given quadratic sequence, we need to examine the pattern of differences between consecutive terms:
The differences between the terms are: 5, (x - 9), (37 - x).
Since it's a quadratic sequence, the differences should have a linear pattern. Thus, we can write the equation:
(x - 9) = 5 + (37 - x)
Simplifying the equation:
x - 9 = 42 - x
Combining like terms:
2x = 51
Dividing both sides by 2:
x = 51 / 2
So, the value of x is 25.5.
b) To determine the nth term of the sequence, we can use the quadratic formula:
an = a + (n - 1)d + (n - 1)(n - 2)c/2
Here, a = 4 (first term),
d = 5 (common difference between terms),
c = (37 - 9) = 28 (coefficient of the quadratic term).
Substituting these values into the formula, we get:
an = 4 + (n - 1)5 + (n - 1)(n - 2)(28/2)
Simplifying further:
an = 4 + 5n - 5 + 14n^2 - 42n + 28
Collecting like terms:
an = 14n^2 - 37n + 27
So, the nth term of the quadratic sequence is 14n^2 - 37n + 27.
To find the missing term and determine the nth term of the given quadratic sequence, we need to identify the pattern and use it to derive the formulas for the missing term and the nth term.
Let's start by analyzing the sequence. The difference between consecutive terms appears to increase with each step.
The first difference between 4 and 9 is 9 - 4 = 5.
The second difference, between 9 and x, is x - 9.
The third difference, between x and 37, is 37 - x.
Since the sequence is quadratic, the second differences must be constant. Therefore, x - 9 = 37 - x.
Solving for x:
2x = 46
x = 23
a) The missing term x is 23.
To determine the nth term of the sequence, we need to establish a general formula. Since the second differences are constant, the sequence follows a quadratic pattern.
We can use the formula for the nth term of a quadratic sequence, which is given as:
T(n) = an² + bn + c
To find the values of a, b, and c, we need to substitute three terms and their respective positions in the sequence into the equation.
Using the first three terms:
T(1) = 4 = a(1)² + b(1) + c
T(2) = 9 = a(2)² + b(2) + c
T(3) = 23 = a(3)² + b(3) + c
Simplifying these equations, we get a system of equations:
a + b + c = 4 (1)
4a + 2b + c = 9 (2)
9a + 3b + c = 23 (3)
Solving this system of equations will give us the values of a, b, and c.
You can use various methods to solve this system, such as substitution or elimination. Here, we will use the elimination method:
First, subtract equation (1) from equation (2) to eliminate c:
(4a + 2b + c) - (a + b + c) = 9 - 4
3a + b = 5 (4)
Next, subtract equation (2) from equation (3) to eliminate c:
(9a + 3b + c) - (4a + 2b + c) = 23 - 9
5a + b = 14 (5)
Now, solve equations (4) and (5) simultaneously:
3a + b = 5 (4)
5a + b = 14 (5)
By subtracting equation (4) from equation (5), we eliminate b:
(5a + b) - (3a + b) = 14 - 5
2a = 9
Dividing by 2:
a = 9/2 = 4.5
Substituting the value of a back into equation (4):
3(4.5) + b = 5
13.5 + b = 5
b = 5 - 13.5
b = -8.5
Substituting the values of a and b back into equation (1):
4.5 + (-8.5) + c = 4
-4 + c = 4
c = 4 + 4
c = 8
Now that we have the values of a, b, and c, we can write the nth term formula:
T(n) = 4.5n² - 8.5n + 8
b) The nth term of the sequence is 4.5n² - 8.5n + 8.
general term for the above:
let term(n) = an^2 + bn + c
for n = 1
4 = a+b+c **
for n = 2
9 = 4a + 2b + c ***
for n = 3
20 = 9a + 3b + c ****
subtract ** from ***
5 = 3a + b ##
subtract *** from ****
11 = 5a + b ###
subtract ## from ###
6 = 2a
a = 3
into ##
5 = 9+b
b = -4
into **
4 = 3 - 4 + c
c = 5
term(n) = 3n^2 - 4n + 5
check for term(4)
LS = 37
RS = 3(16) - 16 + 5 = 37
my answer is correct