Question 1
LEVEL 6
The first four terms of a quadratic sequence are shown below.
2, 11, 22, 35What is the next term in the sequence?
Select the correct answer from the list below.
A: 47
B: 48
C: 49
D: 50
CORRECT ANSWER: D: 50
WORKED SOLUTION:
In a quadratic sequence, the difference between each term will increase by a fixed amount at each step. Looking at the differences, we see
The difference is increasing by 2 at each step, meaning the next difference will be 15, and therefore the next term will be
35 + 15 = 50
Question 2
LEVEL 6
The first four terms of a quadratic sequence are shown below.
8, 15, 28, 47Find the next term in the sequence.
Select the correct answer from the list below.
A: 68
B: 70
C: 72
D: 74
CORRECT ANSWER: C: 72
WORKED SOLUTION:
In a quadratic sequence, the difference between each term will increase by a fixed amount at each step. Looking at the differences, we see
The difference is increasing by 6 at each step, meaning the next difference will be 25, and therefore the next term will be
47 + 25 = 72
Question 3
LEVEL 6
The n^{th} term formula for a quadratic sequence is given by 2n^2 - 6n + 5.
Find the 10^{th} term.
Select the correct answer from the list below.
A: 145
B: 150
C: 155
D: 160
CORRECT ANSWER: A: 145
WORKED SOLUTION:
To find the terms, substitute the values of 10 into the formula, respectively:
u_{10} = 2(10^2) - 6(10) + 5 = 200 - 60 + 5 = 145
Question 4
LEVEL 6
Find the n^{th} term given the following quadratic sequence
1, 6, 17, 34, 57
Select the correct answer from the list below.
A: 3n^2 - 4n + 2
B: 3n^2 + 4n - 2
C: 2n^2 - 4n + 2
D: 3n^2 - 4n - 2
CORRECT ANSWER: A: 3n^2 - 4n + 2
WORKED SOLUTION:
The formula for the nth term of a quadratic sequence will look like
an^2 + bn + c,
where a, b, c are all numbers to be determined. So, we must firstly look at how much our sequence is increasing at each step, and then find the difference between those values. See
To find a we need to divide the difference of the differences by 2. This gives a = 3. We no know the first part is 3n^2.
We then compare the values in the sequence with those generated by 3n^2 and find the difference:
The row of differences, denoted by d, forms a linear sequence, and we must find the nth term formula for that sequence. So, we see
The common difference is -4, so the n^{th} formula begins with -4n.
Now we must check what to add/subtract to -4n to obtain the sequence above. We see that for n = 1.
This gives the correct number as 2.
Answer = u_n = 3n^2 - 4n + 2
Question 5
LEVEL 6
The expression for the nth term of this sequence for 1, 4, 13, 28, 49, can be written in the form an^2-6n + c
Find the value of a and c:
Select the correct answer from the list below.
A: a = 3, c = 4
B: a = 2, c = 2
C: a = 3, c = 2
D: a = 2, c = 4
CORRECT ANSWER: A: a = 3, c = 4
WORKED SOLUTION:
Quadratic sequence is 1, 4, 13, 28, 49
2nd difference = 6, so a = 3;
We know that when n = 1; the term is 1 and when n =2, the term is 4.
So we can write two equations:
For n = 1
1 = 3 × 12 + b + c → \, - 2 = b + c
For n = 2
4 = 3 × 22 + 2b + c → \, - 8 = 2b + c
So b = - 6 and c = 4
Answer = 3n^2 - 6n + 4