**Middle School Number Sense Lesson 87: Adding a Geometric Sequence**

We have covered 5 different sequence addition concepts so far. Today we will discuss adding an

*infinite geometric*sequence. This concept has appeared

**7 times**so far this year, with a median placement at

**question # 61**.

**Like our Facebook Page (**

**https://www.facebook.com/numberdojo/**

**) if you want to see new number sense posts on your wall. I will reward you with a free concept index or flashcard file of your choice. Thank you!**

**Number Dojo Level: 223**

During previous lessons, we have added

*terminating (finite)*sequences, such as:

**1 + 3 + 5 + 7 + 9**, or

**2 + 4 + 6 + 8 + 10**, or

**18 + 15 + 12 + 9 + 6**, or

**1 + 1 + 2 + 3 + 5 + 8 + 13 + 21**, or

**1 + 8 + 27 + 64 + 125**

**16 + 8 + 4 + 2 + 1 + 0.5 + 0.25 + ...**

**one-half**of the previous term.

**1/2**is known as the

**common ratio**of the sequence.

**How to Solve:**

- Note the first term (n) of the sequence.
- Determine the
**common ratio**(r) by dividing any term by the one preceding it. - Use this formula:
**n/(1 - r)**

**Example 1: Find the sum of the infinite geometric series 5 + 2.5 + 1.25 + ... = ___**

- n =
**5** - r = 2.5 ÷ 5 =
**1/2** - n/(1 - r) = 5/(1 - 1/2) = 5/(1/2) = 5 x 2 =
**10**

**Example 2: The infinite geometric series 9 + 3 + 1 ... = ___**

- n =
**9** - r = 3 ÷ 9 =
**1/3** - n/(1 - r) = 9/(1 - 1/3) = 9/(2/3) = 9 x 3/2 =
**27/2**

**Example 3: The sum of the infinite geometric series 8 + 4 + 2 + 1 + ... = ___**

- n =
**8** - r = 4 ÷ 8 =
**1/2** - n/(1 - r) = 8/(1 - 1/2) = 8/(1/2) = 8 x 2 =
**16**

**Example 4: The sum of the infinite geometric series 12 + 3 + 3/4 + ... = ___**

- n =
**12** - r = 3 ÷ 12 =
**1/4** - n/(1 - r) = 12/(1 - 1/4) = 12/(3/4) = 12 x 4/3 =
**16**

**Example 5: The sum of the infinite geometric series, 8 + 6 + 4.5 + ... = ___**

- n =
**8** - r = 6 ÷ 8 =
**3/4** - n/(1 - r) = 8/(1 - 3/4) = 8/(1/4) = 8 x 4 =
**32**

**Example 6: 4/9 - 2/3 + 1 - 3/2 + ... = ___**

*

__: Don't let the alternating signs fool you!__

**Note**- n =
**4/9** - r = (-3/2) ÷ 1 =
**-3/2** - n/(1 - r) = 4/9/(1 - -3/2) = 4/9/(1 + 3/2) = 4/9/(5/2) = 4/9 x 2/5 =
**8/45**

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

addseqgeom.pdf |

**Up Next for Middle School: DiscrimRoot**