Geometric Sequences Revision Notes for Grade 12 NSC Matric Mathematics

Geometric Sequences

What is a geometric sequence?

A geometric sequence is a special type of sequence where each term is found by multiplying the previous term by the same fixed number. This fixed number is called the constant ratio, represented by the letter r.

In simpler terms, if you know one term in a geometric sequence, you can find the next term by multiplying it by r, and the term after that by multiplying by r again, and so on.

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Key characteristics of geometric sequences:

Each term = previous term × constant ratio (r)
The ratio between consecutive terms is always the same
The sequence grows (or shrinks) by multiplication, not addition

Understanding geometric sequences is essential for many mathematical applications, from calculating compound interest to modelling population growth patterns.

Understanding the constant ratio (r)

The constant ratio (r) is the value you multiply each term by to get the next term. To find the constant ratio, divide any term by the term before it.

Formula for constant ratio: $r = \frac{T_2}{T_1} = \frac{T_3}{T_2} = \frac{T_n}{T_{n-1}}$

Where:

$T_1$ = first term
$T_2$ = second term
$T_n$ = nth term
$T_{n-1}$ = term before the nth term

The beauty of geometric sequences lies in this consistency - no matter which consecutive terms you choose, the ratio will always be the same.

Real-world example: Disease spread pattern

Let's look at how a disease might spread to understand geometric sequences better. Imagine each infected person spreads the disease to exactly 2 other people each day.

This creates a pattern where the number of newly infected people doubles each day:

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Worked Example: Analysing Disease Spread Pattern

The sequence of newly infected people is: 2, 4, 8, 16, 32, ...

Let's verify this is geometric by checking the ratios:

$\frac{4}{2} = 2$
$\frac{8}{4} = 2$
$\frac{16}{8} = 2$
$\frac{32}{16} = 2$

Since all ratios equal 2, we confirm this is a geometric sequence with constant ratio r = 2.

This pattern demonstrates how geometric sequences can model exponential growth in real-world situations.

General term formula

For any geometric sequence, we can find a formula to calculate the nth term directly without having to work out all the terms before it.

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The general term formula is: $T_n = ar^{n-1}$

Where:

$T_n$ = the nth term (the term you want to find)
$a$ = the first term of the sequence
$r$ = the constant ratio
$n$ = the position number of the term

Why is the power (n-1)?

Let's examine the pattern step by step:

1st term: $T_1 = a \times r^0 = a \times 1 = a$
2nd term: $T_2 = a \times r^1 = ar$
3rd term: $T_3 = a \times r^2 = ar^2$
nth term: $T_n = a \times r^{n-1}$

The power starts at 0 for the first term, which is why we use $(n-1)$ for the nth term.

How to test if a sequence is geometric

To check whether a sequence is geometric, follow this systematic approach:

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Testing Process:

Calculate the ratio between consecutive terms
Check if all ratios are equal

The test formula is: $\frac{T_2}{T_1} = \frac{T_3}{T_2} = \frac{T_4}{T_3} = \ldots = \text{constant}$

If all ratios are the same, the sequence is geometric. If any ratio is different, it's not geometric.

This verification step is crucial before applying geometric sequence formulas to ensure accuracy in your calculations.

Worked example: Disease spread calculations

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Worked Example: Disease Spread Calculations

Question: Using the flu epidemic example where $T_n = 2 \times 2^{n-1}$ :

Calculate how many newly-infected people there are on the tenth day
On which day will 16,384 people be newly-infected?

Solution:

Step 1: Write down known values

$a = 2$ (first term)
$r = 2$ (constant ratio)
$T_n = 2 \times 2^{n-1}$

Step 2: Calculate $T_{10}$

$T_n = ar^{n-1}$
$T_{10} = 2 \times 2^{10-1}$
$T_{10} = 2 \times 2^9$
$T_{10} = 2 \times 512$
$T_{10} = 1024$

On the tenth day, there are 1024 newly-infected people.

Step 3: Find when $T_n = 16,384$

$T_n = ar^{n-1}$
$16,384 = 2 \times 2^{n-1}$
$16,384 \div 2 = 2^{n-1}$
$8,192 = 2^{n-1}$

Since $8,192 = 2^{13}$ :

$2^{13} = 2^{n-1}$
$13 = n - 1$
$n = 14$

There are 16,384 newly-infected people on the 14th day.

Geometric mean

The geometric mean between two numbers is a value that, when placed between them, creates a geometric sequence.

For a geometric sequence a, x, b: $x = \pm\sqrt{ab}$

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Worked Example: Finding Geometric Mean

Find the geometric mean between 5 and 20.

Step 1: Set up the equation

For the sequence 5, x, 20 to be geometric:

$\frac{x}{5} = \frac{20}{x}$

Step 2: Solve for x

$x^2 = 5 \times 20$
$x^2 = 100$
$x = \pm 10$

Step 3: Check both solutions

Using $x = 10$ : Sequence is 5, 10, 20 with ratio $r = 2$
Using $x = -10$ : Sequence is 5, -10, 20 with ratio $r = -2$

Both are valid geometric sequences.

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Notice that geometric means can produce two different sequences - one with a positive ratio and one with a negative ratio. Both are mathematically correct.

Common exam tips

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Key Points to Remember:

Always check that ratios are constant before using geometric sequence formulas
The general term formula uses $(n-1)$ as the power, not $n$
Geometric means can be positive or negative
When solving for $n$ , use the fact that if $a^x = a^y$ , then $x = y$ (same base rule)
Show all working steps clearly in calculations

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Common mistakes to avoid:

Using $n$ instead of $(n-1)$ in the general term formula
Forgetting to consider both positive and negative geometric means
Not checking if the sequence is actually geometric before applying formulas
Confusing arithmetic sequences (which add a constant) with geometric sequences (which multiply by a constant)

Remember!

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Essential Geometric Sequence Concepts:

Geometric sequences multiply by a constant ratio $r$ to get from one term to the next
Constant ratio $r = \frac{T_n}{T_{n-1}}$ (must be the same for all consecutive terms)
General term formula is $T_n = ar^{n-1}$ where $a$ is the first term
Test for geometric sequence by checking if all consecutive ratios are equal
Geometric mean between $a$ and $b$ is $x = \pm\sqrt{ab}$ , giving two possible sequences

Geometric Sequences (Grade 12 NSC Matric Mathematics): Revision Notes