# Algebra: Number Sequence Word Problems

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

### 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

### 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, _, _, _

### 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:

1. Find the nth term of:
a) 6, 11, 16, 21, 26, …
b) 2, 10, 18, 26, 34, …
c) 8, 6, 4, 2, 0, …

2. 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.
b) 45 is not a term in this number sequence.
Explain why.

3. Here are the first five terms of a number sequence.
3, 9, 15, 21
a) (i) Write down the next term in the sequence.
b) Write down the 7th term in the sequence.
c) Jean says 58 is in the sequence.
Is Jean correct?

### 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 = 2nd difference (always constant)
3a + b = 2nd term - 1st term
a + b + c = 1st term

Examples:

1. Find the nth term Tn of this sequence
3, 10, 21, 36, 55, …
2. Find the nth term Tn of this sequence
0, 7, 20, 39, 64, …

### Sequences - Notation

A sequence is a list of numbers that follow a rule.

Examples:

1. The nth term of a sequence is given by Un = 3n - 1.
Work out:
a. The first term.
b. The third term.
c. The nineteenth term.

2. The nth term of a sequence is given by Un = n2/(n + 1).
Work out:
a. The first three terms.
b. The 49th term.

### Sequences - The nth Term - Given A Term Find n

Examples:

1. A sequence has nth term given by Un = 5n - 2
Find the value of n for which Un = 153

2. A sequence has nth term given by Un = n2 + 5
Find the value of n for which Un = 149

3. A sequence has nth term given by Un = n2 - 7n + 12
Find the value of n for which Un = 72

4. A sequence is generated by the formula Un = 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.

### Sequences - Recurrence Relations

Examples:

1. Find the first four terms of the following sequence
Un + 1 = Un + 4, U1 = 7

2. Find the first four terms of the following sequence
Un + 1 = Un + 4, U1 = 5

3. Find the first four terms of the following sequence
Un + 2 = 3Un + 1 - Un, U1 = 4 and U2 = 2

4. A sequence of terms {Un}, n ≥ 1 is defined by the recurrence relation Un + 2 = mUn, 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.

### Find The nth Term In A Sequence

A method for finding any term in a sequence

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