 # Algebra: Number Sequence Word Problems

Related Topics: More Algebra Word Problems

Number Sequence Problems are word problems that involves a number sequence. Sometimes you may be asked to obtain the value of a particular term of the sequence or you may be asked to determine the pattern of a sequence.

### Number Sequence Problems: Value Of A Particular Term

The question will describe how the sequence of numbers is generated. After a certain number of terms, the sequence will repeat through the same numbers again. Try to follow the description and write down the sequence of numbers until you can determine how many terms before the numbers repeat. That information can then be used to determine what a particular term would be.

For example,

If we have a sequence of numbers:
x, y, z, x, y, z, ....
that repeats after the third term.

If we want to find out what is the fifth term then we get the remainder of 5 ÷ 3, which is 2.

The fifth term is then the same as the second term, which is y.

Example:

The first term in a sequence of number 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, get the remainder for 45 ÷ 4, which is 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:

The fastest way to solve this would be if you notice that the pattern:

6 × 2 + 1 = 13
13 × 2 + 1 = 27

The value of n is 1.

Method 2:

If you were not able to see the pattern then you can come with two equations and then solve for n.

6m + n = 13             (equation 1)
13m + n = 27          (equation 2)

Use substitution method

Isolate n in equation 1

n = 13 – 6m

Substitute 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
How to solve number sequences by looking for patterns, then 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.
(ii) Explain how you worked out your answer.
(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.
(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. 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
Example:
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
Find any term in a sequence

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