Welcome to Series! Your Guide to Patterns in Mathematics (Pure Math 1, Section 1.6)
Hello future mathematician! This chapter on Series is all about discovering and mastering patterns in sequences of numbers. You might think of it as the mathematics of consistent change—whether the change is addition, multiplication, or the growth of polynomial terms.
Why is this important? Series underpin everything from calculating compound interest and modeling radioactive decay to finding the area under curves. Mastering these formulas will save you huge amounts of time in calculations and give you powerful tools for problem-solving in Paper 1!
Section 1: The Binomial Theorem (The Power of Expansion)
The Binomial Theorem provides a systematic way to expand expressions of the form \((a+b)^n\) when $n$ is a positive integer. Instead of multiplying brackets thousands of times, we use a neat formula!
1.1 Key Terminology and Notation
- Binomial: An expression with two terms, like \((a+b)\) or \((2x - 3y)\).
- \(n\): The power to which the binomial is raised. In Pure Mathematics 1, $n$ must be a positive integer (e.g., 3, 5, 10).
- \(\binom{n}{r}\) or \({}^nC_r\): This is the Binomial Coefficient. It tells you the number of ways to choose \(r\) items from a set of \(n\).
- Factorial ($n!$): The product of all positive integers less than or equal to \(n\). Example: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
You can calculate the binomial coefficient using the formula:
$$
\binom{n}{r} = \frac{n!}{r!(n-r)!}
$$
1.2 The General Expansion Formula
The expansion of \((a+b)^n\) is given by:
$$ (a+b)^n = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1} b^1 + \binom{n}{2}a^{n-2} b^2 + \dots + \binom{n}{n}a^0 b^n $$
Don't worry, this looks scarier than it is!
Step-by-Step Trick for the Expansion:
- The power of \(a\) starts at \(n\) and decreases by 1 in each subsequent term.
- The power of \(b\) starts at 0 and increases by 1 in each subsequent term.
- The sum of the powers of \(a\) and \(b\) in every term must always equal \(n\).
- The coefficients are found using \(\binom{n}{r}\).
Example: Expand \((2x - y)^3\). Here \(n=3\), \(a=2x\), and \(b=-y\).
$$ \binom{3}{0}(2x)^3 (-y)^0 + \binom{3}{1}(2x)^2 (-y)^1 + \binom{3}{2}(2x)^1 (-y)^2 + \binom{3}{3}(2x)^0 (-y)^3 $$
Simplifying gives:
$$
1(8x^3)(1) + 3(4x^2)(-y) + 3(2x)(y^2) + 1(1)(-y^3)
$$
Final result: \(8x^3 - 12x^2y + 6xy^2 - y^3\)
Quick Review: Binomial Expansion
Always remember to include the sign when substituting \(b\). If the binomial is \((a-b)\), then \(b\) in the formula should be considered \(-b\).
Section 2: Arithmetic Progressions (APs)
An Arithmetic Progression (AP) is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference ($d$).
Analogy: Think of receiving a fixed, identical raise every year. The increase is constant.
2.1 Identifying Key Components of an AP
- \(a\): The first term.
- \(d\): The common difference. (Found by \(u_2 - u_1\)).
- \(n\): The position or number of the term.
2.2 The \(n\)th Term Formula
To find any term in the sequence (the $n$th term, denoted \(u_n\)), you start at \(a\) and add the difference $d$ exactly \((n-1)\) times.
$$ u_n = a + (n-1)d $$
2.3 Sum of the First \(n\) Terms (\(S_n\))
If you need to add up the first \(n\) terms, you use the sum formula, $S_n$. (This is much faster than manually adding 100 terms!)
The formula found in the MF19 booklet is: $$ S_n = \frac{n}{2}\{2a + (n-1)d\} $$
There is also a handy alternative if you know the last term, $l$: $$ S_n = \frac{n}{2}(a+l) $$
Did you know? The formula for \(S_n\) was supposedly discovered by Carl Friedrich Gauss when he was just a young schoolboy. He quickly found a way to sum the numbers from 1 to 100 by pairing them up (1+100, 2+99, etc.).
2.4 Condition for AP
If three terms, \(a, b, c\), are consecutive terms in an AP, then the difference between them must be equal:
\(b - a = c - b\), which simplifies to:
$$
2b = a + c
$$
Common Mistake to Avoid (AP)
Be careful when finding the difference \(d\). If the sequence is decreasing (e.g., 10, 7, 4...), then \(d\) must be negative! (\(7 - 10 = -3\)).
Key Takeaway for APs: APs are defined by addition of a constant common difference ($d$). The core skill is substituting the correct values for \(a\), \(d\), and \(n\) into the two main formulas, \(u_n\) and \(S_n\).
Section 3: Geometric Progressions (GPs)
A Geometric Progression (GP) is a sequence where the ratio between consecutive terms is constant. This constant ratio is called the common ratio ($r$).
Analogy: Think of compound interest. Your money grows by a fixed percentage (ratio) each year, not a fixed amount.
3.1 Identifying Key Components of a GP
- \(a\): The first term.
- \(r\): The common ratio. (Found by \(u_2 / u_1\)).
- \(n\): The position or number of the term.
3.2 The \(n\)th Term Formula
To find any term in the sequence (the $n$th term, \(u_n\)), you start at \(a\) and multiply by the ratio $r$ exactly \((n-1)\) times.
$$ u_n = ar^{n-1} $$
3.3 Sum of the First \(n\) Terms (\(S_n\))
The sum of the first \(n\) terms of a GP, $S_n$, is given by:
$$ S_n = \frac{a(1-r^n)}{1-r}, \text{ provided } r \neq 1 $$
Why are there two forms of the sum formula?
Sometimes you might see the formula written as: $$ S_n = \frac{a(r^n-1)}{r-1} $$
These are mathematically identical! We usually use the first version (with \(1-r\)) when \(|r|<1\) to keep the denominator positive, and the second version (with \(r-1\)) when \(|r|>1\) for the same reason. Choose the one that makes your calculations easiest.
3.4 Condition for GP
If three terms, \(a, b, c\), are consecutive terms in a GP, then the ratio between them must be equal:
\(b / a = c / b\), which simplifies to:
$$
b^2 = ac
$$
Key Takeaway for GPs: GPs are defined by multiplication by a constant common ratio ($r$). The core skill is substituting the correct values for \(a\), \(r\), and \(n\) into the formulas, \(u_n\) and \(S_n\).
Section 4: Sum to Infinity (\(S_{\infty}\))
This is one of the most interesting concepts in series. Can you add up an infinite number of terms and get a finite answer? Yes, but only under one very strict condition!
4.1 The Condition for Convergence
A GP is said to converge (meaning its sum approaches a fixed, finite value as \(n\) tends to infinity) if and only if its common ratio \(r\) satisfies the condition:
$$ |r| < 1 \quad \text{or} \quad -1 < r < 1 $$
If \(r\) is less than 1 (but greater than -1), each subsequent term gets smaller and smaller, heading towards zero. Eventually, adding more terms makes almost no difference to the total sum.
Imagine cutting a cake in half, then cutting the remainder in half, and so on forever. You will never exceed the size of the original cake. The total volume you "sum" converges to the volume of the original cake.
4.2 The Sum to Infinity Formula
If the convergence condition \(|r| < 1\) is met, we use the formula for the Sum to Infinity, \(S_{\infty}\):
$$ S_{\infty} = \frac{a}{1-r} $$
Crucial Check!
NEVER use the \(S_{\infty}\) formula unless you have first verified that \(-1 < r < 1\). If \(|r| \ge 1\), the series diverges and the sum to infinity is undefined (or infinitely large).
Key Takeaway for \(S_{\infty}\): Convergence only happens if \(|r| < 1\). If it does converge, the sum is simply determined by the first term ($a$) and the common ratio ($r$).
Review Box: Formulas You Must Know (and Use!)
Arithmetic Progression (AP)
- \(n\)th term: \(u_n = a + (n-1)d\)
- Sum of \(n\) terms: \(S_n = \frac{n}{2}\{2a + (n-1)d\}\)
Geometric Progression (GP)
- \(n\)th term: \(u_n = ar^{n-1}\)
- Sum of \(n\) terms: \(S_n = \frac{a(1-r^n)}{1-r}\)
- Sum to Infinity (Convergent only, \(|r|<1\)): \(S_{\infty} = \frac{a}{1-r}\)
Binomial Expansion (\(n\) positive integer)
- General Term \((r+1)\)th term: \(T_{r+1} = \binom{n}{r} a^{n-r} b^r\)
Practice solving problems that mix these concepts—for example, finding the common ratio of a GP given the sum to infinity, or finding the number of terms in an AP required to exceed a certain total. You've got this!