Wize High School Grade 11 Math Textbook > Sequences & Series

Geometric Series

Geometric Series

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

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

$\boxed{\quad S_n = \dfrac{a(r^n-1)}{r-1}, \quad r\ne 1 \quad}$

Note: an equivalent formula can be written using the first term, $t_1$ , and the $(n+1)$ th term, $t_{n+1}$ :

$\boxed{\quad S_n = \dfrac{t_{n+1}-t_1}{r-1}, \quad r\ne 1 \quad}$

Example

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

What is the associated geometric series? ________

What is the common ratio? $r=$ ________

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

The first term is $a=2$ , and we want to find the sum of the first $n=4$ terms:

$\begin{aligned} S_n &= \dfrac{a(r^n-1)}{r-1}\\[1em] S_4 &= \dfrac{2(3^4-1)}{3-1} = \dfrac{2(81-1)}{2} = \boxed{80} \end{aligned}$

When $n$ is small, we can easily check: $2+6+18+54 = 80 \quad \colorThree{\checkmark}$

Example: Geometric Series

Would you rather:

receive one million dollars ($1 000 000) today, or
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 $n=30$ days.

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

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

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

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

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.

A) $1000+500+250+125+\dots$

B) $5,25,125,625,\ \dots$

C) $-1+1-1+1-1+\dots$

D) $0.01+0.1+1+10+\dots$

E) $4+8+12+16+\dots$

I don't know

Practice: Geometric Series

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

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

a) the sum of this series isb) the number of terms n is

I don't know

Practice: Geometric Series

Illustration for practice question: 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?

Daily average time she held her breath (to the nearest second):

I don't know

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

$\boxed{\quad S_n = \dfrac{a(r^n-1)}{r-1}}$

where,

$S_n$ is the nth partial sum
$a$ is the first term in the geometric sequence
$r$ 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

$\boxed{\quad S_n = \dfrac{t_{n+1}-t_1}{r-1}}$

where,

$S_n$ is the nth partial sum
$t_{n+1}$ (or $a_{n+1}$ ) is the n+1 term in the geometric sequence
$t_1$ (or $a_1$ ) is the first term in the geometric sequence
$r$ 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)

Geometric Series

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