In these lessons, we will look at solving word problems involving number sequences.
Related Pages
Number Sequences
Linear Sequences
Geometric Sequences
Quadratic and Cubic Sequences
Number Sequence Problems are word problems that involve generating and using number sequences. Sometimes you may be asked to obtain the value of a particular term of a sequence or you may be asked to determine the pattern of a sequence.
A number sequence problem may first describe how a sequence of numbers is generated. After a certain number of terms, the sequence will repeat. Follow the description of the sequence and write down numbers in sequence until you can determine how many terms occur before the numbers repeat. Then use that information to determine what a particular term could be.
For example:
If we have a sequence of numbers:
x, y, z, x, y, z, ….
that repeats after the third term, to find the fifth term we find the remainder of 5 divided
by 3, which is 2. (5 ÷ 3 is 1 remainder 2).
The fifth term is then the same as the second term, which is y.
Example:
The first term in a sequence of numbers is 2. Each even-numbered term is 3 more than the
previous term and each odd-numbered term, excluding the first, is –1 times the previous
term. What is the 45th term of the sequence?
Solution:
Step 1: Write down the terms until you notice a repetition.
2, 5, -5, - 2, 2, 5, -5, -2, …
The sequence repeats after the fourth term. Step 2: To find the 45th term, find the remainder for 45 divided by 4, which is 1. (45 ÷ 4 is 11 remainder 1)
Step 3: The 45th term is the same as the 1st term, which is 2.
Answer: The 45th term is 2.
Example:
6, 13, 27, 55, …
In the sequence above, each term after the first is determined by multiplying the preceding term by m and then adding n. What is the value of n?
Solution:
Method 1:
Notice the pattern:
6 × 2 + 1 = 13
13 × 2 + 1 = 27
The value of n is 1.
Method 2:
Write the description of the sequence as two equations with the unknowns m and n, as shown
below, and then solve for n.
6m + n = 13 (equation 1)
13m + n = 27 (equation 2)
Using the substitution method
Isolate n in equation 1
n = 13 – 6m
Substitute n = 13 – 6m into equation 2
13m + 13 – 6m = 27
7m = 14
m = 2
Substitute m = 2 into equation 1
6(2) + n = 13
n = 1
Answer: n = 1
This is a method to solve number sequences by looking for patterns, followed by using addition,
subtraction, multiplication, or division to complete the sequence.
Step 1: Look for a pattern between the given numbers.
Step 2: Decide whether to use +, -, × or ÷
Step 3: Use the pattern to solve the sequence.
Examples:
2, 5, 8, 11, _, _, _
2, 4, 8, 16, _, _, _
15, 12, 9, _, _, _
48, 24, 12, _, _, _
Examples:
Find the nth term of:
a) 6, 11, 16, 21, 26, …
b) 2, 10, 18, 26, 34, …
c) 8, 6, 4, 2, 0, …
Here are the first five terms of a number sequence.
2, 7, 12, 17, 22
a) (i) Write down the next term in the sequence.
(ii) Explain how you worked out your answer.
b) 45 is not a term in this number sequence.
Explain why.
Here are the first five terms of a number sequence.
3, 9, 15, 21
a) (i) Write down the next term in the sequence.
(ii) Explain how you worked out your answer.
b) Write down the 7th term in the sequence.
c) Jean says 58 is in the sequence.
Is Jean correct?
You must give a reason for your answer.
When trying to find the nth term of a quadratic sequence, it will be of the form:
an^{2} + bn + c
where a, b, c always satisfy the following equations
2a = 2^{nd} difference (always constant)
3a + b = 2^{nd} term - 1^{st} term
a + b + c = 1^{st} term
Examples:
A sequence is a list of numbers that follow a rule.
Examples:
The nth term of a sequence is given by U_{n} = 3n - 1.
Work out:
a. The first term.
b. The third term.
c. The nineteenth term.
The nth term of a sequence is given by U_{n} = n^{2}/(n + 1).
Work out:
a. The first three terms.
b. The 49th term.
Examples:
A sequence has nth term given by U_{n} = 5n - 2
Find the value of n for which U_{n} = 153
A sequence has nth term given by U_{n} = n^{2} + 5
Find the value of n for which U_{n} = 149
A sequence has nth term given by U_{n} = n^{2} - 7n + 12
Find the value of n for which U_{n} = 72
A sequence is generated by the formula U_{n} = an + b where a and b are constants to be found. Given that U_{3} = 5 and U_{8} = 20 find the values of the constants a and b.
Examples:
Find the first four terms of the following sequence
U_{n + 1} = U_{n} + 4, U_{1} = 7
Find the first four terms of the following sequence
U_{n + 1} = U_{n} + 4, U_{1} = 5
Find the first four terms of the following sequence
U_{n + 2} = 3U_{n + 1} - U_{n}, U_{1} = 4 and U_{2} = 2
A sequence of terms {U_{n}}, n ≥ 1 is defined by the recurrence relation U_{n + 2} = mU_{n}, where m is a constant. Given also U_{1} = 2 and U_{2} = 5.
a. find an expression in terms of m for U_{3}.
b. find an expression in terms of m for U_{4}.
Given the value of U_{4} = 21:
c. find the possible values of m.
A method for finding any term in a sequence
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