Grade 11

Advanced

9 min

Lesson 8 of 12

Geometric Sequences and Series

Not Started

Explore sequences with constant ratios and determine the behavior of infinite sums.

If you could fold a standard piece of paper 42 times, would you believe it would reach the moon? This explosive growth is the secret power of geometric sequences, where a simple constant ratio transforms the small into the astronomical.

Learning Objectives

1.Calculate the nth term of any geometric sequence using the general formula
2.Find the sum of finite geometric series using the ratio-based summation formula
3.Determine if an infinite series converges and calculate its sum when $|r| < 1$

The DNA of Geometric Sequences

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio ( $r$ ). Unlike arithmetic sequences that add a constant, geometric sequences grow or shrink exponentially. To find the common ratio, simply divide any term by its predecessor: $r = \frac{a_n}{a_{n-1}}$ . The general formula to find the $n$ th term is $a_n = a_1 \cdot r^{n-1}$ , where $a_1$ is the first term and $n$ is the position of the term. This formula allows us to 'jump' to any point in the sequence without calculating every step in between.

Find the 10th term of the sequence: $3, 6, 12, 24, ...$

1. Identify the first term: $a_1 = 3$ . 2. Find the common ratio: $r = \frac{6}{3} = 2$ . 3. Use the formula $a_n = a_1 \cdot r^{n-1}$ for $n=10$ . 4. Substitute the values: $a_{10} = 3 \cdot 2^{10-1} = 3 \cdot 2^9$ . 5. Calculate: $a_{10} = 3 \cdot 512 = 1536$ .

Quick Check

If a geometric sequence starts with $10$ and has a common ratio of $0.5$ , what is the third term?

Multiply 10 by 0.5 twice, or use the formula with

n=3

Answer

2.5

Summing the Finite

When we add the terms of a geometric sequence, we create a geometric series. Calculating the sum of the first

n

terms manually is tedious, so we use the finite sum formula:

S_n = \frac{a_1(1 - r^n)}{1 - r}

This formula is incredibly powerful for financial modeling, such as calculating the total value of an investment over time where interest is compounded. Note that

r

cannot equal

1

, as that would result in division by zero (in that case, the sum is simply

n \cdot a_1

Find the sum of the first 6 terms of the series: $100 + 50 + 25 + ...$

1. Identify $a_1 = 100$ , $r = 0.5$ , and $n = 6$ . 2. Plug into the formula: $S_6 = \frac{100(1 - 0.5^6)}{1 - 0.5}$ . 3. Simplify the denominator: $1 - 0.5 = 0.5$ . 4. Calculate the power: $0.5^6 = 0.015625$ . 5. Solve: $S_6 = \frac{100(0.984375)}{0.5} = \frac{98.4375}{0.5} = 196.875$ .

Quick Check

In the finite sum formula, what happens to the sum as $n$ increases if $r > 1$ ?

Think about what happens to

r^n

when

r

is a large number.

Answer

The sum grows toward infinity (diverges).

The Paradox of Infinite Sums

Can you add an infinite number of things and get a finite result? Yes! If the common ratio is a fraction between

-1

and

1

(i.e.,

|r| < 1

), the terms get smaller and smaller, eventually approaching zero. We say the series converges. In this case, as

n

approaches infinity, the

r^n

term in our sum formula vanishes, leaving us with the infinite sum formula:

S = \frac{a_1}{1 - r}

|r| \geq 1

, the terms do not shrink, and the sum diverges, meaning it has no finite limit.

A ball is dropped from $10$ meters. Each time it hits the ground, it bounces back to $\frac{2}{3}$ of its previous height. Find the total vertical distance the ball travels before coming to rest.

1. The downward distances are $10, 10(\frac{2}{3}), 10(\frac{2}{3})^2...$ 2. The upward distances are $10(\frac{2}{3}), 10(\frac{2}{3})^2...$ 3. This is two infinite series. Let's sum the downward path first: $a_1 = 10, r = \frac{2}{3}$ . 4. $S_{down} = \frac{10}{1 - 2/3} = \frac{10}{1/3} = 30$ . 5. The upward path starts at $10(\frac{2}{3}) = 6.67$ : $S_{up} = \frac{20/3}{1 - 2/3} = 20$ . 6. Total distance = $30 + 20 = 50$ meters.

Key Takeaways

1A geometric sequence follows the rule $a_n = a_1 \cdot r^{n-1}$ .
2The sum of a finite geometric series is $S_n = \frac{a_1(1 - r^n)}{1 - r}$ .
3An infinite series converges only if the absolute value of the ratio is less than 1 ( $|r| < 1$ ).
4The sum of a convergent infinite series is $S = \frac{a_1}{1 - r}$ .

Quick Check

Multiple Choice

What is the common ratio $r$ for the sequence $8, -4, 2, -1, ...$ ?

Multiple Choice

Which of the following infinite series converges?

True/False

If $|r| = 1$ , the infinite sum formula $S = \frac{a_1}{1-r}$ can still be used.

What's Next?

Review Tomorrow

In 24 hours, try to write down the infinite sum formula and the specific condition required for a series to converge.

Practice Activity

Look at a repeating decimal like $0.333...$ and try to express it as an infinite geometric series where $a_1 = 0.3$ and $r = 0.1$ to prove it equals $1/3$ .

Browse

Browse

Geometric Sequences and Series

Learning Objectives

The DNA of Geometric Sequences

Summing the Finite

The Paradox of Infinite Sums

Key Takeaways

Quick Check

What's Next?

Geometric Sequences and Series

Learning Objectives

The DNA of Geometric Sequences

Summing the Finite

The Paradox of Infinite Sums

Key Takeaways

Quick Check

What's Next?