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.

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.

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*.

6

m+n=13 (equation 1)

13m+n= 27 (equation 2)

Use substitution method

Isolate *n* in equation 1

n =13 – 6m

Substitute into equation 2

13

m+ 13 – 6m= 27

7m= 14

m= 2

Substitute *m* = 2 into equation 1

6(2) +

n= 13

n= 1

Answer: *n* = 1

Math Made Easy: Solving Number Sequences

How to solve number sequences by looking for patterns, then using addition, subtraction, multiplication, or division to complete the sequence.

Number Sequences

The relationship between the sequence of squares and the sequence of odd numbers.

Find the nth term rule from a sequence.

Find any term in a sequence

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