# Geometric Sequences

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## Objective

SWBAT find the nth partial sum of a geometric series.

#### Big Idea

Dive into some work with geometric sequences to gain understanding and prove a formula.

## Check Homework

10 minutes

One question from yesterday’s homework that students always ask about is #4. In this video I go through one strategy that usually seems to make sense for students.

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## Launch and Explore

20 minutes

Going into this lesson, my students have an understanding of what a geometric sequence is and how to describe one explicitly and recursively. Today we are going to focus on finding the nth term and the nth partial sum of a geometric series.

I start class by giving them this worksheet and have them go through questions #1-7 for about 15 minutes or so with their table groups. These problems are more challenging than what they have been working on because they must figure out bothr explicit and recursive formulas given the second and fourth term, for example.

One conceptual hurdle that I notice in my students' work is when the common ratio is not a rational number. For Problem #4, for example, the second term is 5 and the fourth term is 50. If a student gets stuck on this, I ask him/her to write the sequence like 5, ____ , 50 with a blank missing for the third term. Then I ask him/her to consider how many times to multiply by the common ratio to get from 5 to 50. Once they realize it is twice, they can usually set up the equation 5x2 = 50 and then solve for x.

For Question #7, my students don’t usually recall the formula, so they use their graphing calculator. This is good! I also want them to realize that they can’t use the Rainbow Method that we used for arithmetic series. In the next section we will work on finding a formula for the nth partial sum of a geometric series.

If there are any problems that gave students a lot of trouble from #1 to #6, we will address them as a class before concluding this section of the lesson. However, during this lesson, students usually clear up any confusion with their group or with me as I am moving around the room.

## Extend

15 minutes

I always start by asking if any student remembers the formula for the nth partial sum of a geometric series. My students may not know the formula but they may remember that it involves a fraction and exponents. I will ask them what we would need to know about the geometric series in order to find the nth partial sum and they will usually decide on the first term, the common ratio, and the last term or the term number. I tell them that we will definitely need these things as we work on a formula for the nth partial sum.

In the video below I explain my approach to proving the formula for the nth partial sum of a geometric series.

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After going through the proof we will use it to verify the partial sums from Question #7 on the worksheet. To end the lesson, we recap the formula for the nth partial sum for a geometric and arithmetic series. It’s really important that students understand when to use each tool and are using them appropriately.