**Middle School Number Sense Lesson 56: Finding the Nth Term in a Sequence**

Last week we figured out how to determine the next term in a given sequence. Today we will cover finding

*any*term in a sequence. This concept appeared

**5 times**on middle school tests last year, with a median placement at

**question # 56**.

**Number Dojo Level: 249**

There are at least three important skills that are needed to master this concept:

- Identifying the pattern in the sequence,
- Manipulating the numbers to make the calculation of the Nth term easier, and
- Calculating the Nth term.

**Example 1: The 16th term of the sequence 3, 7, 11, ... is ___**

- Each term is 4 more than the one before it. In other words, the terms are spaced
**4 apart**. **Add 1**to the 1st term to make it equal to the spacing. 3 + 1 =**4**.- Multiply this by the term # we are finding. 4 x 16 =
**64**. Then**subtract the 1**we added in Step 2. 64 - 1 =**63**. This is our answer.

**Example 2: The 80th term of the sequence 7, 12, 17, ... is ___**

- The spacing between terms is
**5**. **Subtract 2**from the 1st term to make it equal the spacing. 7 - 2 =**5**.- Multiply this by the term # we are finding. 5 x 80 =
**400**. Then**add back the 2**we subtracted in Step 2. 400 + 2 =**402**.

**Example 3: How many terms does the sequence 2, 5, 8, 11, ..., 152 have?**

- The spacing between terms is
**3**. **Add 1**to the 1st term to equal the spacing. 2 + 1 =**3**. This makes the last term 152 + 1 =**153**.- Divide this by the spacing to determine the number of terms. 153 ÷ 3 =
**51**. This is our answer.

**Example 4: The 8th term of the arithmetic sequence, 11, 8, 5, 2, ... is ___**

- The spacing between terms is
**-3**. **Subtract 14**from the 1st term to equal the spacing. 11 - 14 =**-3**.- Multiply this by the term # we are finding. -3 x 8 =
**-24**. Then**add back the 14**we subtracted in Step 2. -24 + 14 =**-10**.

**Example 5: How many terms are in the sequence 7, 16, 25, 34, 43, ..., 223? ___**

- The spacing is
**9**. **Add 2**to the 1st term to equal the spacing. 7 + 2 =**9**. This makes the last term 223 + 2 =**225**.- Divide this by the spacing to determine the number of terms. 225 ÷ 9 =
**25**. This is our answer.

**Example 6: Find the 7th term of the Fibonacci sequence, 1, 1, 2, 3, ... ___**

- Each term of the Fibonacci sequence is the sum of the two before it.
- We are given the first 4 terms. Simply determine each of the next 3 terms, in order, keeping track of the # of terms along the way.
- The 5th term is 2 + 3 =
**5**. The 6th term is 3 + 5 =**8**. The 7th term is 5 + 8 =**13**.

**Here's a free worksheet to help you practice SeqNthTerm:**

seqnthterm.pdf |

**Up Next for Middle School: Circles**