Arithmetic & Geometric Sequences: Your Ultimate Study Guide!
Hey everyone! Welcome to your study notes for one of the most interesting topics in algebra: sequences and series. Don't worry if this sounds complicated; it's all about recognising and working with patterns in numbers. You see patterns everywhere, from the petals on a flower to your savings account balance.
In this chapter, we'll explore two main types of sequences:
- Arithmetic Sequences: Where you add the same amount each time. (Think: consistent, steady change).
- Geometric Sequences: Where you multiply by the same amount each time. (Think: rapid growth or decay).
Understanding these will help you predict future values, calculate totals over time, and tackle a whole range of real-life problems. Let's get started!
Part 1: Arithmetic Sequences (AS)
What is an Arithmetic Sequence?
An Arithmetic Sequence (or AS) is a list of numbers where the difference between any two consecutive terms is constant. It's that simple! We're just adding or subtracting the same number over and over again.
Analogy: Imagine you're climbing a staircase where every single step has the exact same height. The height of each step is your constant difference!
We have two key players in any AS:
- The First Term (a): This is where the sequence begins. It's our starting point.
- The Common Difference (d): This is the fixed number we add to get to the next term. It can be positive, negative, or even a fraction!
To find the common difference, just subtract any term from the term that comes after it. For example, `Term 2 - Term 1`.
Examples:
- Sequence: 3, 7, 11, 15, ...
First Term (a) = 3
Common Difference (d) = 7 - 3 = 4. We add 4 each time. - Sequence: 20, 18, 16, 14, ...
First Term (a) = 20
Common Difference (d) = 18 - 20 = -2. We add -2 (or subtract 2) each time.
The General Term Formula: Finding any term (T_n)
What if you need to find the 50th term? You don't want to write out 50 numbers! That's why we have a formula for the general term, often called `T_n` or `T(n)`, which represents the term in the n-th position.
Let's figure it out logically:
- 1st Term: `T(1) = a`
- 2nd Term: `T(2) = a + d`
- 3rd Term: `T(3) = a + d + d = a + 2d`
- 4th Term: `T(4) = a + 3d`
See the pattern? For the n-th term, we add the common difference `(n-1)` times. This gives us the most important formula for AS:
The General Term Formula (AS)
$$ T(n) = a + (n-1)d $$Step-by-Step Example:
Find the 30th term of the sequence 5, 8, 11, ...
- Identify a and d.
The first term is 5, so `a = 5`.
The common difference is 8 - 5 = 3, so `d = 3`. - Identify n.
We want the 30th term, so `n = 30`. - Plug into the formula.
`T(30) = a + (30-1)d`
`T(30) = 5 + (29)(3)`
`T(30) = 5 + 87`
`T(30) = 92`
So, the 30th term is 92. Easy, right?
Sum of an Arithmetic Sequence: Finding the total (S_n)
Sometimes we need to add up the terms in a sequence. This is called a summation or a series. The sum of the first `n` terms is called `S_n` or `S(n)`.
Did you know?
A famous mathematician, Carl Friedrich Gauss, supposedly figured out how to sum the numbers from 1 to 100 in seconds when he was just a child! He paired the first and last numbers (1+100=101), the second and second-to-last (2+99=101), and realised there were 50 such pairs. 50 × 101 = 5050. This is the logic behind our first sum formula!
We have two handy formulas for this. Which one you use depends on what information you have.
The Sum Formulas (AS)
1. If you know the first and last terms:
$$ S_n = \frac{n}{2}(a + l) $$where `l` is the last term, `T(n)`.
2. If you know the first term and common difference (most common):
$$ S_n = \frac{n}{2}[2a + (n-1)d] $$Step-by-Step Example:
Find the sum of the first 20 terms of the sequence 2, 6, 10, 14, ...
- Identify a, d, and n.
`a = 2`
`d = 6 - 2 = 4`
`n = 20` - Choose the right formula.
We have `a`, `d`, and `n`, so the second formula is perfect. - Plug into the formula.
`S(20) = \frac{20}{2}[2(2) + (20-1)(4)]`
`S(20) = 10[4 + (19)(4)]`
`S(20) = 10[4 + 76]`
`S(20) = 10[80]`
`S(20) = 800`
The sum of the first 20 terms is 800.
Key Takeaway: Arithmetic Sequences
An AS involves adding a constant value (`d`) repeatedly.
- To find any term: `$$ T(n) = a + (n-1)d $$`
- To find the sum of terms: `$$ S_n = \frac{n}{2}[2a + (n-1)d] $$`
Part 2: Geometric Sequences (GS)
What is a Geometric Sequence?
A Geometric Sequence (or GS) is a list of numbers where you get to the next term by multiplying by a constant value. This value can make the sequence grow very fast or shrink towards zero.
Analogy: Imagine a special bouncing ball. After each bounce, it only reaches a certain percentage (say, 80%) of its previous height. That percentage is the constant multiplier!
The two key players in a GS are:
- The First Term (a): Same as before, it's our starting value.
- The Common Ratio (r): This is the fixed number we multiply by to get the next term.
To find the common ratio, just divide any term by the term that came before it. For example, `Term 2 / Term 1`.
Examples:
- Sequence: 3, 6, 12, 24, ...
First Term (a) = 3
Common Ratio (r) = 6 / 3 = 2. We multiply by 2 each time. - Sequence: 100, 50, 25, 12.5, ...
First Term (a) = 100
Common Ratio (r) = 50 / 100 = 0.5. We multiply by 0.5 (or divide by 2) each time.
The General Term Formula: Finding any term (T_n)
Just like with AS, we need a formula to find any term `T(n)` without writing them all out. Let's look at the pattern:
- 1st Term: `T(1) = a`
- 2nd Term: `T(2) = a \times r = ar^1`
- 3rd Term: `T(3) = a \times r \times r = ar^2`
- 4th Term: `T(4) = ar^3`
The pattern is that the power of `r` is always one less than the term number `n`. This gives us our general term formula for GS:
The General Term Formula (GS)
$$ T(n) = ar^{n-1} $$Common Mistake Alert!
The formula is `ar^(n-1)`, NOT `(ar)^(n-1)`. Remember your order of operations (BIDMAS/PEDMAS) - you must calculate the power part (`r^(n-1)`) first, then multiply by `a`.
Step-by-Step Example:
Find the 7th term of the sequence 5, 15, 45, ...
- Identify a and r.
The first term is 5, so `a = 5`.
The common ratio is 15 / 5 = 3, so `r = 3`. - Identify n.
We want the 7th term, so `n = 7`. - Plug into the formula.
`T(7) = a r^(7-1)`
`T(7) = 5 \times 3^6`
`T(7) = 5 \times 729`
`T(7) = 3645`
So, the 7th term is 3645.
Sum of a Finite Geometric Sequence (S_n)
Now, let's add up the first `n` terms of a GS. This is super useful for problems involving compound interest or investments over a set period.
The Sum Formula (GS)
There is one main formula, but we write it in two ways to make calculations easier (especially to avoid negative numbers).
$$ S_n = \frac{a(r^n - 1)}{r - 1} \quad \text{or} \quad S_n = \frac{a(1 - r^n)}{1 - r} $$Pro Tip: Use the first version when `|r| > 1` and the second when `|r| < 1`. They give the same answer!
Step-by-Step Example:
Find the sum of the first 8 terms of the sequence 2, 8, 32, ...
- Identify a, r, and n.
`a = 2`
`r = 8 / 2 = 4`
`n = 8` - Choose the best formula version.
Since `r = 4` (which is > 1), let's use the first version to keep the denominator positive. - Plug into the formula.
`S(8) = \frac{2(4^8 - 1)}{4 - 1}`
`S(8) = \frac{2(65536 - 1)}{3}`
`S(8) = \frac{2(65535)}{3}`
`S(8) = \frac{131070}{3}`
`S(8) = 43690`
The sum of the first 8 terms is 43690.
Sum to Infinity: Adding forever (S_∞)
This is a really cool concept. What happens if you add the terms of a GS... forever? You might think the sum would be infinite. But for certain sequences, the sum is actually a finite number!
Analogy: Imagine walking towards a wall. First you walk half the distance. Then you walk half of the remaining distance. Then half of that, and so on. You will get infinitely close to the wall, but the total distance you walk will never be more than the initial distance to the wall. Your total distance "converges" to a single value.
This only works under one very important condition:
The condition for a sum to infinity to exist is `$$ -1 < r < 1 $$` (or `$$|r| < 1$$`).
If the common ratio `r` is between -1 and 1 (but not 0), each term gets smaller and smaller, so the sum approaches a fixed limit.
The Sum to Infinity Formula
When the condition is met, the formula is surprisingly simple:
$$ S_{\infty} = \frac{a}{1 - r} $$Step-by-Step Example:
Find the sum to infinity of the sequence 18, 6, 2, ...
- Identify a and r.
`a = 18`
`r = 6 / 18 = 1/3` - Check the condition.
Is `$$ -1 < r < 1 $$`? Yes, `$$ -1 < 1/3 < 1 $$`. So, the sum to infinity exists. - Plug into the formula.
`S_{\infty} = \frac{a}{1 - r}`
`S_{\infty} = \frac{18}{1 - 1/3}`
`S_{\infty} = \frac{18}{2/3}`
`S_{\infty} = 18 \times \frac{3}{2}`
`S_{\infty} = 27`
If you were to add all the terms of this sequence forever, the total would be exactly 27.
Key Takeaway: Geometric Sequences
A GS involves multiplying by a constant value (`r`) repeatedly.
- To find any term: `$$ T(n) = ar^{n-1} $$`
- To find the sum of n terms: `$$ S_n = \frac{a(r^n - 1)}{r - 1} $$`
- To find the sum to infinity (only if `|r|<1`): `$$ S_{\infty} = \frac{a}{1 - r} $$`
Part 3: Real-Life Problems
This is where it all comes together! The key is to read the question carefully and decide whether the situation describes an Arithmetic or a Geometric sequence.
- Is a fixed amount being added or subtracted? -> Arithmetic
- Is the value changing by a fixed percentage or multiplier? -> Geometric
Example 1: Salary Plan (Arithmetic)
Connie starts a job with an annual salary of $300,000. Her company promises a fixed annual raise of $15,000 every year. What will be her total earnings after working for 8 years?
Thinking Process:
- A "fixed annual raise" means we are ADDING the same amount each year. This is an AS.
- Identify the key values: `a = 300000`, `d = 15000`.
- We want "total earnings" after 8 years, so we need the SUM, `S_n`, with `n = 8`.
- Use the sum formula for AS: `S_n = \frac{n}{2}[2a + (n-1)d]`
- `S(8) = \frac{8}{2}[2(300000) + (8-1)(15000)]`
- `S(8) = 4[600000 + (7)(15000)]`
- `S(8) = 4[600000 + 105000]`
- `S(8) = 4[705000] = 2820000`
Answer: Connie's total earnings after 8 years will be $2,820,000.
Example 2: Car Depreciation (Geometric)
A new car costs $250,000. Its value depreciates by 20% each year. What is the value of the car after 5 years?
Thinking Process:
- "Depreciates by 20%" means the value is MULTIPLIED by a percentage each year. This is a GS.
- Identify the key values: `a = 250000`.
- Be careful with `r`! If the value loses 20%, it keeps 80% of its value. So, `r = 1 - 0.20 = 0.80`.
- We want the value "after 5 years". This is a common tricky point. `T(1)` is the initial value (after 0 years). `T(2)` is the value after 1 year. So, the value after 5 years is the 6th term, `T(6)`. Therefore, `n = 6`.
- Use the general term formula for GS: `T(n) = ar^(n-1)`
- `T(6) = 250000 \times (0.8)^{6-1}`
- `T(6) = 250000 \times (0.8)^5`
- `T(6) = 250000 \times 0.32768 = 81920`
Answer: The value of the car after 5 years is $81,920.
And that's a wrap! The key to mastering sequences is practice. Read the question, identify the type of sequence, pull out your `a`, `d` or `r`, and then choose the right formula. You've got this!