**Middle School Number Sense Lesson 55: Finding the Next Term in a Sequence**

We have already covered three topics where we added numbers in a sequence (in

**AddSequence**,

**AddSeqOdd**, and

**AddSeqEven**). Today we will learn to see a pattern in the sequence and determine the next term. This concept appeared

**8 times**last year, with a median placement at

**question # 53**.

**Number Dojo Level: 221**

There are several different types of questions that fall into this category. Sometimes the pattern in the sequence is obvious, and other times it takes a bit more deciphering. I will try to cover the most common variations in the examples.

**Example 1: The next term of the arithmetic sequence ... 2 1/4, 3/4, - 3/4, - 2 1/4, ... is ___**

- The spacing between each two terms in this sequence is 1 1/2. To find the next term, subtract 1 1/2 from the last term given.
- - 2 1/4 - 1 1/2 =
**- 3 3/4**.

**Example 2: The next term in the sequence 0, 1, 8, 27, ... is ___**

- In this sequence, we need to recognize that each term is a perfect cube (of 0, 1, 2, and 3, respectively).
- The next term is 4 x 4 x 4, which is
**64**.

**Example 3: The next term in the sequence 3, 6, 4, 7, 5, ... is ___**

- Notice that every other term increases by 1. (Or you may have noticed that the spacing is +3, -2, +3, -2).
- The next term is (7 + 1) or (5 + 3), which is
**8**.

**Example 4: The next term in the sequence 3, 6, 11, 18, ... is ___**

- In this sequence, the spacing between the consecutive numbers increases by 2 each time (so the spacing is 3, then 5, then 7).
- To find the next term, and 9 to 18 and get
**27**.

**Example 5: The next term of 1, 1, 2, 3, 5, ... is ___**

- In this sequence, each term is the sum of the two terms preceeding it. This represents the classic
**Fibonacci Sequence**. - To find the next term, add the last 2 terms. 3 + 5 =
**8**.

**Example 6: In the arithmetic sequence ..., x, 14, y, 16, z, ..., the value of x + y + z is ___**

- It is important to understand that the spacing in an arithmetic sequence is determined by adding (not multiplying).
- We can quickly recognize that y is halfway between 14 and 16, so it must be
**15**. - That makes the spacing between the terms 1, so x must be
**13**and z must be**17**. x + y + z = 13 + 15 + 17 =**45**. - OR, we could tap into our understanding of adding sequences (
). There are 3 terms in the sequence x, y, z, and the median is 15. So 3 x 15 =*remember the # of terms times the median?***45**.

**Example 7: If w, x, 4, y, z form a geometric sequence, wxyz = ___**

- It's important to understand that the spacing in a geometric sequence is determined by multiplying (not adding).
- We could choose a specific ratio to form the spacing, and establish a geometric sequence that would use that ratio. Let's say the ratio is 2 (so each term is twice the previous one). That would make
**y = 8**and**z = 16**. Then**x = 2**and**w = 1**. So wxyz = (1)(2)(8)(16) = 16 x 16 =**256**. - OR, we could set our unknown ratio to be
**a**. Then x would be**4/a**, and y would be**4a**. So xy would have to be (4/a)(4a). The a's cancel, and we're left with 4 x 4 =**16**. Similarly, w would be**4/aa**, and z would be**4aa**. So wz would have to be (4/aa)(4aa). The aa's cancel, and we're left with 4 x 4 =**16**. wxyz = (wz)(xy) = (16 x 16) =**256**.

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

seqnextterm.pdf |

**Up Next for Middle School: SeqNthTerm**