AS Level
Mathematics 9709 P1
6. Series
Written by: Tharun Athreya
Formatted by: Dhyaneshwaran V
Index
6.1 Binomial Expansion
- ‘Binomial’ means ‘two terms’.
- The binomial theorem gives the expansion of the expressions in the form \( (a + b)^n \)
where \( n \)
is a positive integer. - The expansion involves the sum of terms featuring binomial coefficient.
🔥 Binomial Theorem General Formula
- \[ (a + b)^n = a^n + \frac{n!}{1!(n – 1)!} a^{n-1} b + \frac{n!}{2!(n – 2)!} a^{n-2} b^2 + \frac{n!}{3!(n – 3)!} a^{n-3} b^3 + \dots \]
- Where \( \frac{n!}{r!(n – r)!} \) represents the combination formula.
-
You can use Pascal’s triangle instead of \( \frac{n!}{r!(n – r)!} \)
- Pascal’s triangle can be used for small \( n \) values. For larger \( n \) values, it is prone to arithmetic errors and it will take time.
-
Pascal’s Triangle:
🔥 Questions asked in the exam:
- Asked to expand the brackets for first few terms
- Asked to find a coefficient of a particular term
- Solving problems in unknowns
6.2 Arithmetic Progressions
- A linear sequence such as 5, 8, 11, 14, 17, … is called an arithmetic progression.
- Each term differs from the term before by a constant, where the constant is called the common differentiation.
- The common difference is also allowed to be zero or negative.
- \( \text{nth term} = a + (n – 1) \cdot d \)
- \( a \) is the first term
- \(d \) is the common difference
🔥 Exam Tips
- The formula will be given in the data booklet
- If you know two terms in the sequence in an arithmetic progression, you can find a and d using simultaneous equations.
- When the terms in the sequence are added together we call the resulting sum a series.
- The sum of an arithmetic progression, \( S_n \)
, can be written as:
\( S_n = \frac{n}{2} \cdot \left( a + l \right) \) or \( S_n = \frac{n}{2} \cdot \left( 2a + (n – 1) \cdot d \right) \)
- \(l \) is the last term
- \(a \) is the first term
- \(d \) is the common difference
- You can use any formula according to the question.
- How do I derive the formula for the sum of an arithmetic progression?
- Learn this proof of the sum of an arithmetic progression formula – you can be asked to give it on the exam:
- Write the terms out once in order.
- Write the terms out again in reverse order.
- Add the two sums together:
- The terms will pair up to give the same sum \( 2a + (n – 1) \cdot d \)
- There will be n of these terms
- Divide by two as two of the sums have been added together
- The terms will pair up to give the same sum \( 2a + (n – 1) \cdot d \)
- Learn this proof of the sum of an arithmetic progression formula – you can be asked to give it on the exam:
🔥 Exam Tip
- The arithmetic series formulas are in the formulae booklet – you don’t need to memorise them.
- To find out the \( \text{nth term} \)
using the sum of an arithmetic progression,\[
\text{nth term} = S_n – S_{n-1}
\]
6.3 Geometric Progressions
- In a geometric progression (also called geometric sequence) there is a common ratio between consecutive terms in the sequence
- \( \text{nth term} = a \cdot r^{(n-1)} \)
- \( a \) is the first term
- \( r \) is the common ratio
🔥 Exam Tip
- The formula is given in the formula booklet
- If you know two terms in a geometric sequence you can find \( a \) and \( r \) using simultaneous equations.
- The sum of the terms of a geometric progression is sometimes called a geometric series
- The sum of a geometric progression, \( S_n \)
, can be written as:
\( S_n = \frac{a(1 – r^n)}{1 – r} \) or \( S_n = \frac{a(r^n – 1)}{r – 1} \)
Either formula can be used but it is usually easier to:
- Use the first formula when \( -1 < r < 1 \)
- Use the second formula when \( r > 1 \)
or when \( r \leq -1 \) - How do I prove the formula for the sum of a geometric progression?
- Learn this proof of the sum of a geometric progression formula – you can be asked to give it in the exam:
- Write out the sum once
- Write out the sum again but multiply each term by r
- Subtract the second sum from the first
- All the terms except the two should cancel out
- Factorise and rearrange to make S the subject
- Learn this proof of the sum of a geometric progression formula – you can be asked to give it in the exam:
6.4 Infinite Geometric series
- An infinite sequence is a sequence whose terms continue forever.
- If (and only if) \( |r| < 1 \)
, then the sum of a geometric progression converges to a finite value given by the formula - \( S_{\infty} = \frac{a}{1 – r} \) provided that \( -1 < r < 1 \)
- If \( |r| \geq 1 \)
the sum of a geometric progression is divergent and the sum to infinity does not exist.
6.5 Further arithmetic and geometric series
- There are many real- life situations which can be modelled using progressions.
- If a quantity is changing repeatedly by having a fixed amount added to or subtracted from it then the use of arithmetic progressions is appropriate.
- If a quantity is changing repeatedly by a fixed percentage, or by being multiplied repeatedly by a fixed amount, then the use of geometric progressions is appropriate.
- Exam Tips: To help you do this, be suspicious of questions about savings accounts, salaries, sales commissions, profits and the like – these are often progression questions in disguise!