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Wize High School Grade 11 Math Textbook > Sequences & Series

Geometric Series

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Geometric Series

A geometric series is the sum of the terms of a geometric sequence.

Wize Concept
Recall, for a geometric sequence:
  • aa is the first term (also called t1t_1)
  • rr is the common ratio (found by dividing any term by the term that comes before it)

The sum of the first nn terms of a geometric series (called the partial sum) is given by:

Sn=a(rn−1)r−1,r≠1\boxed{\quad S_n = \dfrac{a(r^n-1)}{r-1}, \quad r\ne 1 \quad}

Note: an equivalent formula can be written using the first term, t1t_1, and the (n+1)(n+1)th term, tn+1t_{n+1}:

Sn=tn+1−t1r−1,r≠1\boxed{\quad S_n = \dfrac{t_{n+1}-t_1}{r-1}, \quad r\ne 1 \quad}
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Example

Consider the geometric sequence 2,6,18,54, …2,6,18,54,\ \dots

What is the associated geometric series?
2+6+18+54+...

What is the common ratio? r=r=
3

What is the sum of the first 4 terms of this series?

The first term is a=2a=2, and we want to find the sum of the first n=4n=4 terms:
Sn=a(rn−1)r−1S4=2(34−1)3−1=2(81−1)2=80\begin{aligned} S_n &= \dfrac{a(r^n-1)}{r-1}\\[1em] S_4 &= \dfrac{2(3^4-1)}{3-1} = \dfrac{2(81-1)}{2} = \boxed{80} \end{aligned}

When nn is small, we can easily check: 2+6+18+54=80✓2+6+18+54 = 80 \quad \colorThree{\checkmark}
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Example: Geometric Series

Would you rather:
  1. receive one million dollars ($1 000 000) today, or
  2. receive a penny ($0.01) today, and then double that tomorrow, and double again the next day, and so on for 30 days?
Let's use geometric series to calculate the total amount of money received in the second scenario, where n=30n=30 days.

In this scenario, the geometric series is: 0.01+0.02+0.04+0.08+0.16+⋯+t300.01+0.02+0.04+0.08+0.16+\dots+t_{30}

The first term of the series corresponds to the amount on the first day: t1=a=0.01t_1 =a =0.01

The common ratio is how much we multiply the amount by every day: r=2r=2

Sn=a(rn−1)r−1S30=0.01(230−1)2−1=0.01(230−1)Note: 230=1 073 741 824=$ 10 737 418.23\begin{aligned} S_n &= \dfrac{a(r^n-1)}{r-1}\\[1em] S_{30} &= \dfrac{0.01(2^{30}-1)}{2-1}\\[1em] &= 0.01(2^{30}-1) \quad \text{Note: }2^{30} = 1\,073\,741\,824 \\[0.5em] &= \$\,10\,737\,418.23 \end{aligned}

Therefore, by waiting 30 days, the second scenario allows you to receive more than 10 times what is offered in the first scenario!

This is the power of geometric growth, which is the same exponential growth but with discrete steps (not continuous).

Practice: Geometric Series

Select all of the options that are geometric series.

Practice: Geometric Series

a) Determine the sum of the geometric series 1600+800+400+⋯+251600+800+400+\dots+25.

b) Determine the number of terms in the geometric series −2+6−18+⋯+tn-2+6-18+\dots+t_n whose sum is 3280.

Practice: Geometric Series


A diver wants to practice holding her breath once per day. On her first day of training, she holds her breath for 30 seconds. Every day, she manages to hold her breath 10% longer than the day before.

Over these two weeks, what was the average amount of time she managed to hold her breath every day?

Summary - Geometric Series

A geometric series is when we sum up (add up) all the terms in a geometric sequence.

The nth partial sum of a geometric series is when we add up the first n numbers in the geometric sequence. There are two formulas that calculate the nth partial sum.

Formula #1

Sn=a(rn−1)r−1\boxed{\quad S_n = \dfrac{a(r^n-1)}{r-1}}
where,
  • SnS_n is the nth partial sum
  • aa is the first term in the geometric sequence
  • rr is the common ratio between terms in the geometric sequence (we get this by dividing any number in the geometric sequence by the previous number)

Formula #2

Sn=tn+1−t1r−1\boxed{\quad S_n = \dfrac{t_{n+1}-t_1}{r-1}}
where,
  • SnS_n is the nth partial sum
  • tn+1t_{n+1} (or an+1a_{n+1}) is the n+1 term in the geometric sequence
  • t1t_1 (or a1a_1) is the first term in the geometric sequence
  • rr is the common ratio between terms in the geometric sequence (we get this by dividing any number in the geometric sequence by the previous number)