Algebra: Number Sequence Word Problems

In these lessons, we will look at solving word problems involving number sequences.

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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.

Number Sequence Problems: Value Of A Particular Term

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.

Number Sequence Problems: Determine The Pattern Of A Sequence

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

Solving Number Sequences

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, _, _, _

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Sequences - Find The nth Term

Describe a linear sequence
Find the next few terms
Find the nth term
Use the nth term to find a term in the sequence

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.

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Find The nth Term Of A Quadratic Sequence

When trying to find the nth term of a quadratic sequence, it will be of the form:

an² + 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:

Find the n^th term T_n of this sequence
3, 10, 21, 36, 55, …
Find the n^th term T_n of this sequence
0, 7, 20, 39, 64, …

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Sequences - Notation

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 = n²/(n + 1).
Work out:
a. The first three terms.
b. The 49th term.

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Sequences - The nth Term - Given A Term Find n

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 = n² + 5
Find the value of n for which U_n = 149
A sequence has nth term given by U_n = n² - 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 U₃ = 5 and U₈ = 20 find the values of the constants a and b.

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Sequences - Recurrence Relations

Examples:

Find the first four terms of the following sequence
U_{n + 1} = U_n + 4, U₁ = 7
Find the first four terms of the following sequence
U_{n + 1} = U_n + 4, U₁ = 5
Find the first four terms of the following sequence
U_{n + 2} = 3U_{n + 1} - U_n, U₁ = 4 and U₂ = 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 U₁ = 2 and U₂ = 5.
a. find an expression in terms of m for U₃.
b. find an expression in terms of m for U₄.
Given the value of U₄ = 21:
c. find the possible values of m.

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Find The nth Term In A Sequence

A method for finding any term in a sequence

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