Cogito
Algebra 2 · Chapter 7 · Lesson 3
Infinite Geometric Series
Adding forever and still landing on a number.
12 problems · about 21 minutes · A-SSE.B.4
What this lesson teaches
The student determines whether an infinite geometric series converges and finds its sum.
- An infinite geometric series converges exactly when |r| < 1.
- Its sum is a₁ ÷ (1 − r).
- Every repeating decimal is one of these series in disguise.
Warm Up
Straightforward practice. Get the method working first.
5 problemsFirst term 10, ratio 0.5. What is the infinite sum?
Answer 20
Why 20.
When does an infinite geometric series converge?
Answer When the size of the ratio is below 1.
Why When |r| < 1.
First term 4, ratio 0.5. Infinite sum?
Answer 8
Why 4 ÷ 0.5.
First term 9, ratio 0.5. Infinite sum?
Answer 18
Why 9 ÷ 0.5.
First term 1, ratio 0.8. Infinite sum?
Answer 5
Why 1 ÷ 0.2.
Build It Up
The same ideas with more to keep track of.
3 problemsFirst term 3, ratio 0.25. Infinite sum?
Answer 4
Why 3 ÷ 0.75.
Ratio 2. Does the series converge? Enter 1 for yes or 0 for no.
Answer 0
Why The terms grow.
0.5 + 0.05 + 0.005 + ... What is the sum, as a decimal to three places?
Answer 0.556
Why 0.5 ÷ 0.9.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsWhich ratios give a convergent infinite series?
Answer r = 0.3; r = −0.5
Why The size must be under 1, and the sign does not matter.
First term 20, ratio 0.5. Infinite sum?
Answer 40
Why 20 ÷ 0.5.
The Total: First term 6, ratio 0.5. What is the infinite sum?
Answer 12
Why 12.
The Bouncing Ball: A ball is dropped from 10 m and each bounce reaches 0.5 of the previous height. What is the total of all the bounce heights after the drop, in metres?
Answer 10 m
Why 10 m.