4.1.1
Pascal's Triangle
What is Pascal's Triangle?
What is Pascal's Triangle?
Pascal's triangle is an arrangement of numbers that allow us to quickly work out the coefficients of terms when you expand expressions of the form .
How do you construct Pascal's triangle?
How do you construct Pascal's triangle?
- To construct Pascal's triangle, we start by writing a 1.
- In the row below, we write two 1’s either side.
- For the third row, we add the two 1's in the second row to give 2, and write this in between them on the row below.
- The start and ending numbers of a row are always equal to 1.
Pascal's triangle
Pascal's triangle
- We continue this way, adding two numbers to find the number in the row below it until we have written out as many rows as we would like.
Expanding binomials
Expanding binomials
- We can use the numbers in each row of a pascal triangle to expand expressions of the form .
- The first row is equal to .
Expanding binomials
Expanding binomials
- The second row gives the coefficients of the terms of the expansion.
- The coefficients of the terms of the expansion are given by the th row of Pascal's triangle.
Expanding binomials
Expanding binomials
- Each term in the expansion has a total power that is equal to the power of the expansion.
- For example,
- Each term contains powers that add up to three.
Example
Example
- What is the expansion of ?
Write out Pascal's triangle up to row 5
Write out Pascal's triangle up to row 5
- The coefficients of the terms in the expansion are given by the th row of Pascal's triangle.
Write out coefficients of expansion
Write out coefficients of expansion
- The coefficients are:
Multiply coefficients
Multiply coefficients
- Multiply each coefficient by decreasing orders of and increasing orders of
:
Write the sum of the terms
Write the sum of the terms
- The answer is the sum of each term:
Answer
Answer
- We can simplify our answer by working out the powers:
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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