4,9,x,37 is a quadratic sequence (a)show that x=20 (b)determine the nth term of the sequence thanks in advance
quadratic sequences will have linear 1st differences and constant 2nd differences.
The 1st differences here are 5,x-9,37-x
The 2nd differences are x-14,46-2x
If x=20, that makes the 1sts: 5,11,17
and the 2nds 6,6
which is consistent.
I expect you can determine the nth term all right...
4du
To show that x = 20 in the quadratic sequence 4, 9, x, 37, we need to find the formula for the nth term of the sequence first.
(a) To find x = 20, we can use the quadratic formula for the nth term of a quadratic sequence, which is given by:
nth term = an^2 + bn + c,
where a, b, and c are constants.
We can substitute the values of the known terms into the formula to create a system of equations:
When n = 1, nth term = 4:
4 = a(1)^2 + b(1) + c -> Equation 1
When n = 2, nth term = 9:
9 = a(2)^2 + b(2) + c -> Equation 2
When n = 4, nth term = 37:
37 = a(4)^2 + b(4) + c -> Equation 3
Now, let's solve the system of equations to find the values of a, b, and c.
From equation 1:
4 = a + b + c -> Equation 4
From equation 2:
9 = 4a + 2b + c -> Equation 5
From equation 3:
37 = 16a + 4b + c -> Equation 6
Next, we can solve this system of equations simultaneously. Subtract Eq 4 from Eq 5:
(9 - 4) = (4a + 2b + c) - (a + b + c)
5 = 3a + b -> Equation 7
Subtract Eq 4 from Eq 6:
(37 - 4) = (16a + 4b + c) - (a + b + c)
33 = 15a + 3b -> Equation 8
Now we have two equations with two variables (Equations 7 and 8). We can solve this system of equations to find the values of a and b.
Multiply Eq 7 by 3 and Eq 8 by 5 to eliminate b:
15 = 9a + 3b -> Equation 9
165 = 75a + 15b -> Equation 10
Subtract Eq 9 from Eq 10:
(165 - 15) = (75a + 15b) - (9a + 3b)
150 = 66a
a = 150/66
a = 25/11
Now substitute the value of a into Eq 9 to find b:
15 = (9 * 25/11) + 3b
165/11 - 225/11 = 3b
-60/11 = 3b
b = -20/11
Finally, we substitute the values of a and b into Eq 4 to find c:
4 = 25/11 - 20/11 + c
44/11 - 20/11 = c
c = 24/11
So the equation for the nth term of the quadratic sequence is:
nth term = (25/11)n^2 - (20/11)n + (24/11)
Now we can substitute n = 3 into the equation to find x:
x = (25/11)(3)^2 - (20/11)(3) + (24/11)
x = 75/11 - 60/11 + 24/11
x = 39/11
x = 3.54
Therefore, x is not equal to 20 in the given quadratic sequence.
(b) To determine the nth term of the quadratic sequence, we have already found the equation above:
nth term = (25/11)n^2 - (20/11)n + (24/11)
To solve for x in the given quadratic sequence, we need to find the pattern and use it to determine the missing term. Let's analyze the sequence step-by-step:
Step 1: Find the common difference:
To determine the common difference, we subtract each term from the previous term.
9 - 4 = 5
x - 9 = x - 9
37 - x = 37 - x
Since the common difference remains constant, we can set the above equations equal to each other:
5 = 5
x - 9 = 5
37 - x = x - 9
Step 2: Solve for x:
Taking the second equation, we have:
x - 9 = 5
x = 5 + 9
x = 14
Taking the third equation, we have:
37 - x = x - 9
37 + 9 = x + x
46 = 2x
x = 46/2
x = 23
Since we obtained two different values for x, we should revisit the problem to check if there may have been a mistake or if it is not a quadratic sequence.