π β’ Module 5 : Sequences, Graphs and Shapes Flashcards
Understand and apply arithmetic, geometric sequences, graphing techniques, and key geometric principles like Pythagoras, averages, and similar shapes.
What is an arithmetic (linear) sequence?
A sequence where each term is generated by adding or subtracting a constant value to the previous term.
Example : 2, 5, 8, 11, 14 is a sequence where 3 is added each time.
How do you find the nth term of an arithmetic sequence?
Use the formula aβ = aβ + (n - 1) x d
Where aβ is the nth term, aβ is the first term, and d is the common difference.
Example : For the sequence 3, 7, 11, the nth term is aβ = 3 + (n - 1) x 4
What is the Fibonacci sequence?
A sequence where each term is the sum of the two preceding terms.
Example : 0, 1, 1, 2, 3, 5, 8, 13, 21β¦
What is a rational number?
A number that can be expressed as a fraction,
p/q, where p and q are integers, and q β 0
Example 3/4, 5, -2
What is an irrational number?
A number that cannot be expressed as a fraction of two integers. It has a non-repeating, non-terminating decimal expansion.
Example : Ο, β2, e.
How do you convert a recurring decimal to a fraction?
Let the decimal be π₯. Multiply by 10, 100, etcβ¦ to create an equation, then subtract the original equation from the new one to eliminate the decimal.
Example : π₯ = 0.333β¦, multiply by 10: 10π₯ = 3.333β¦. Subtract the equations: 10π₯ - π₯ = 3.333β¦-0.333β¦, so 9π₯ = 3, hence π₯=1/3.
How do you plot a straight-line graph from an equation?
Use the equation y=mπ₯+c
Where m is the gradient (slope), and c is the y-intercept.
Plot the y-intercept, then use the gradient to find other points.
What is the gradient of a straight-line graph?
The gradient (or slope) of a straight line is the ratio of the vertical change to the horizontal change between two points on the line.
Formula for gradient : (π¦β - π¦β)/(π₯β - π₯β)
How do you find the equation of a line given two points?
Use the formula for the gradient, m = (π¦β - π¦β)/(π₯β - π₯β), and then use the point-slope form of the equation of a line:
π¦-π¦β=m(π₯-π₯β)
Substitute the values for m, π₯β, and π¦β into the equation to find the equation of the line.
What is the x-intercept of a line?
The x-intercept of a line is the point where the line crosses the x-axis. At this point, y = 0.
What is the y-intercept of a line?
the y-intercept of a line is the point where the line crosses the y-axis. At this point, x = 0.
How do you find the length of a line segment between two points on a graph?
The length of a line segment between two points, (π₯β, π¦β) and (π₯β, π¦β) is found using the distance formula:
d = β(π₯β-π₯β) Β² + (π¦β-π¦β)Β²
What is the midpoint of a line segment?
The midpoint of a line segment between two points, (π₯β, π¦β) and (π₯β, π¦β), is given by:
Midpoint = ((π₯β+ π₯β)/2), (π¦β+ π¦β)/2))
What is the Pythagorean Theorem?
The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b):
aΒ² + bΒ²=cΒ²
How do you apply the Pythagorean Theorem to find the length of a side?
To find the length of a missing side, rearrange the Pythagorean Theorem :
c = βaΒ²+ bΒ²
a = βcΒ²-bΒ²
β¦depending on which side is missing.
What are similar triangles?
Similar triangles have the same shape but may have different sizes. Their corresponding angles are equal, and their corresponding sides are proportional.
How do you calculate the mean (average) of a set of numbers?
The mean is calculated by adding all the numbers together and dividing the sum by the number of numbers.
Mean = Sum of all values/Number of values
How do you calculate the median of a set of numbers?
The median is the middle value in a set of numbers when they are arranged in order.
If there are an even number of values, the median is the average of the two middle values.
How do you calculate the mode of a set of numbers?
The mode is the number that appears most frequently in a set of numbers.
If no number repeats, there is no mode.
How do you calculate the range of a set of numbers?
The range is calculated by subtracting the smallest value from the largest value in a set.
Range = Largest value - smallest value
