IB Mathematics AA SL Geometric sequences and series Study Notes
IB Mathematics AA SL Geometric sequences and series Study Notes Study Notes Offer a clear explanation of Geometric sequences and series , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Geometric sequences and series
Geometric Sequences and Series
Geometric Sequences and Series
A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio \( r \).
General form:
\( a, ar, ar^2, ar^3, \dots \)
- \( a \) is the first term
- \( r \) is the common ratio
n-th term of a geometric sequence:
\( u_n = ar^{n-1} \)
A geometric series is what you get when you add the terms of a geometric sequence. This sum is called a geometric series.
Sum of first \( n \) terms:
If \( r \neq 1 \):
\( S_n = a \dfrac{1 – r^n}{1 – r} \)
Sum to infinity (when \( |r| < 1 \)):
\( S_\infty = \dfrac{a}{1 – r} \)
Note: A geometric series converges only when \( |r| < 1 \).
Example
Given a geometric sequence with first term \( a = 5 \) and common ratio \( r = 2 \):
- Find the 6th term.
- Find the sum of the first 6 terms.
▶️ Answer/Explanation
Solution:
- 6th term:
\( u_6 = ar^{5} = 5 \cdot 2^5 = 5 \cdot 32 = \rm{160} \) - Sum of first 6 terms:
\( S_6 = a \dfrac{1 – r^6}{1 – r} = 5 \cdot \dfrac{1 – 2^6}{1 – 2} = 5 \cdot \dfrac{1 – 64}{-1} = 5 \cdot 63 = \rm{315} \)
Use of Sigma Notation for Sums of Geometric Sequences
Use of Sigma Notation for Sums of Geometric Sequences
Sigma notation is a concise way to write the sum of the terms of a sequence. For a geometric sequence, it is written as:
\( \sum_{k=0}^{n-1} ar^k \)
- \( a \): first term
- \( r \): common ratio
- \( n \): number of terms
- \( k \): index of summation, usually starting at 0 or 1
Sum formula: For a geometric sequence with common ratio \( r \neq 1 \), the sum of the first \( n \) terms is:
\( S_n = a \dfrac{1 – r^n}{1 – r} \)
Example
Use sigma notation to express and find the sum of the first 5 terms of the geometric sequence with \( a = 3 \), \( r = 2 \).
▶️ Answer/Explanation
Solution:
The sequence is: \( 3, 6, 12, 24, 48 \)
Write the sum in sigma notation:
\( \sum_{k=0}^{4} 3 \cdot 2^k \)
Apply the geometric sum formula:
\( S_5 = 3 \cdot \dfrac{1 – 2^5}{1 – 2} = 3 \cdot \dfrac{1 – 32}{-1} = 3 \cdot 31 = \rm{93} \)
Spreadsheets, GDCs, and Graphing Software for Sequences
Spreadsheets, GDCs, and Graphing Software for Sequences
Technological tools like spreadsheets (Excel, Google Sheets), Graphic Display Calculators (e.g., TI-84), and graphing software (e.g., Desmos, GeoGebra) can be used to generate and display sequences efficiently. They are especially useful for visualizing patterns and calculating many terms quickly.
Example 1: Arithmetic Sequence in Excel
Generate the sequence: \( u_n = 5 + 3(n – 1) \), for \( n = 1 \) to \( 10 \)
▶️ Answer/Explanation
- In column A, type numbers 1 through 10.
- In column B, enter the formula for the sequence:
=5 + 3*(A1 - 1) - Drag the formula down to fill all 10 rows.
- You will get the terms: 5, 8, 11, 14, 17, 20, 23, 26, 29, 32
Example 2: Geometric Sequence on GDC
Plot the sequence \( u_n = 2 \cdot 3^{n-1} \) using a TI-84 calculator.
▶️ Answer/Explanation
- Press
MODEand chooseSEQ(sequence mode). - Go to
Y=and enter:u(n) = 2 * 3^(n - 1) - Set the window:
nMin = 1,nMax = 10 - Press
GRAPHto view the sequence plotted as dots.
Example 3: Recursive Sequence in Desmos
Define the recursive sequence: \( u_1 = 1 \), \( u_n = 0.5 u_{n-1} + 3 \)
▶️ Answer/Explanation
- Open Desmos
- Enter:
a[1] = 1 - Enter:
a[n] = 0.5a[n-1] + 3 - Type
table([a[n] for n = 1...20])to see the values. - Use the graph panel to visualize the convergence.
Applications of Sequences and Series
Applications of Sequences and Series
Arithmetic and geometric sequences can model real-life scenarios such as the spread of disease, salary increases/decreases, and population growth or decay. The type of sequence depends on whether the change is constant (arithmetic) or proportional (geometric).
Example 1: Spread of Disease
A virus spreads such that the number of infected people triples every day. If 5 people are infected on day 1, how many people are infected on day 7?
▶️ Answer/Explanation
This is a geometric sequence: \( u_n = u_1 \cdot r^{n-1} \)
Here, \( u_1 = 5 \), \( r = 3 \), \( n = 7 \)
\( u_7 = 5 \cdot 3^{6} = 5 \cdot 729 = 3645 \)
Answer: \( \rm{3645} \) people infected on day 7
Example 2: Salary Increase
An employee earns $\$30,000$ in the first year, and receives a $\$2000$ raise every year. What is their salary in the 5th year?
▶️ Answer/Explanation
This is an arithmetic sequence: \( u_n = u_1 + (n – 1)d \)
\( u_1 = 30000 \), \( d = 2000 \), \( n = 5 \)
\( u_5 = 30000 + (5 – 1)\cdot 2000 = 30000 + 8000 = 38000 \)
Answer: \( \rm{\$38,000} \) in year 5
Example 3: Population Decrease
A population of 10,000 is declining at a rate of 5% per year. What will the population be after 4 years?
▶️ Answer/Explanation
This is geometric decay: \( u_n = u_1 \cdot r^{n-1} \)
\( u_1 = 10000 \), \( r = 0.95 \), \( n = 5 \)
\( u_5 = 10000 \cdot 0.95^4 \approx 10000 \cdot 0.8145 = 8145 \)
Answer: \( \rm{8145} \) people after 4 years