Introduction to Geometric Series
Welcome to the world of Geometric Series! In your previous studies, you might have looked at Arithmetic Series where we add the same number every time. In a Geometric Series, the secret is multiplication. These series appear everywhere—from the way interest grows in a bank account to the path of a bouncing ball. Don't worry if it sounds complex; once you understand the pattern of the "common ratio," the rest is just following a simple map.
What is a Geometric Series?
A geometric sequence is a list of numbers where each term is found by multiplying the previous one by a fixed, non-zero number called the common ratio (\(r\)). When we add these terms together, it becomes a Geometric Series.
Key Terms to Know:
\(a\) = The first term of the series.
\(r\) = The common ratio (the number we multiply by).
\(n\) = The number of terms we are looking at.
\(u_n\) = The value of the \(n\)-th term.
Example: In the series \(3 + 6 + 12 + 24 + ...\)
The first term \(a = 3\).
To get from 3 to 6, we multiply by 2. To get from 6 to 12, we multiply by 2. So, the common ratio \(r = 2\).
How to find \(r\): Simply divide any term by the term before it: \(r = \frac{u_2}{u_1}\) or \(r = \frac{u_3}{u_2}\).
The General Term (\(n\)-th term)
If you want to find a specific term in the series without writing the whole thing out, you can use the general term formula. This is a formula you must remember as it is not provided on the formula sheet.
The Formula: \(u_n = a r^{n-1}\)
Why \(n-1\)? Think about it this way: to get to the 2nd term, you multiply the 1st term by \(r\) once. To get to the 3rd term, you multiply by \(r\) twice. So, to get to the \(n\)-th term, you multiply by \(r\) exactly \(n-1\) times!
Example: Find the 10th term of the series where \(a = 5\) and \(r = 3\).
\(u_{10} = 5 \times 3^{(10-1)}\)
\(u_{10} = 5 \times 3^9 = 98,415\)
The Sum of the First \(n\) Terms (\(S_n\))
Sometimes we need to know the total of all the terms added together. This is called the sum to \(n\) terms. This formula is provided on your formula sheet, so you don't need to memorize it, but you must know how to use it.
The Formula: \(S_n = \frac{a(1 - r^n)}{1 - r}\)
Top Tip: If \(r\) is greater than 1, some students prefer to use \(S_n = \frac{a(r^n - 1)}{r - 1}\) to avoid dealing with negative numbers. Both versions give the exact same answer!
Step-by-Step Example:
Find the sum of the first 6 terms of \(2 + 10 + 50 + ...\)
1. Identify \(a\): \(a = 2\).
2. Identify \(r\): \(10 \div 2 = 5\), so \(r = 5\).
3. Identify \(n\): We want the sum of 6 terms, so \(n = 6\).
4. Plug into the formula: \(S_6 = \frac{2(1 - 5^6)}{1 - 5}\)
5. Calculate: \(S_6 = \frac{2(1 - 15625)}{-4} = \frac{-31248}{-4} = 7,812\)
The Sum to Infinity (\(S_{\infty}\))
This is one of the coolest concepts in math! If a series goes on forever, you might think the sum would always be infinity. However, if the terms keep getting smaller and smaller, the total sum actually settles down to a specific number. We call this a convergent series.
The Condition for Convergence:
A geometric series only has a sum to infinity if the common ratio \(r\) is between -1 and 1. We write this as \(|r| < 1\).
The Formula (Provided on formula sheet):
\(S_{\infty} = \frac{a}{1 - r}\)
Did you know? Imagine you are standing 2 meters from a wall. You move 1 meter toward it (half the distance). Then you move 0.5 meters (half again), then 0.25 meters... You will keep moving forever, but you will never actually pass the wall! Your total distance is \(1 + 0.5 + 0.25 + ...\) which equals exactly 2 meters.
Example: Find the sum to infinity of \(10 + 5 + 2.5 + 1.25 + ...\)
1. \(a = 10\).
2. \(r = 0.5\) (Since \(5 \div 10 = 0.5\)).
3. Check condition: \(|0.5| < 1\), so we can find \(S_{\infty}\).
4. \(S_{\infty} = \frac{10}{1 - 0.5} = \frac{10}{0.5} = 20\).
Sigma Notation (\(\sum\))
You may see geometric series written using the Greek letter Sigma (\(\sum\)). This is just a shorthand way of writing "the sum of."
For example: \(\sum_{r=1}^{n} a r^{r-1}\) means "Add up all the terms from the 1st term to the \(n\)-th term."
Note: Don't confuse the variable \(r\) often used in sigma notation with the common ratio \(r\); always look at the context of the formula!
Common Pitfalls to Avoid
1. Using the wrong formula: Always double-check if the question asks for the term (\(u_n\)) or the sum (\(S_n\)).
2. Negative Ratios: If \(r\) is negative (e.g., \(-0.5\)), the terms will alternate between positive and negative. Be very careful with brackets when calculating \(r^n\). For example, \((-0.5)^2 = 0.25\), but \((-0.5)^3 = -0.125\).
3. Power of \(n-1\): Remember that the formula for the term uses \(n-1\), but the formula for the sum uses \(n\).
4. Sum to infinity validity: If a question asks "Explain why this series has a sum to infinity," you must state that \(|r| < 1\) and show your calculated value for \(r\).
Key Takeaways Summary
Geometric Series: terms found by multiplying by \(r\).
\(n\)-th term: \(u_n = a r^{n-1}\) (Memory check!).
Sum of \(n\) terms: \(S_n = \frac{a(1 - r^n)}{1 - r}\) (On formula sheet).
Sum to infinity: \(S_{\infty} = \frac{a}{1 - r}\) (On formula sheet).
Validity: \(S_{\infty}\) only exists if \(-1 < r < 1\).