Sequences and Series Flashcards
Prove sum of arithmetic series formula
Set to Sn in one direction:
a + (a+d) + (a+2d) … (a+(n-2)d) + (a+(n-1)d)
Set up in other direction
Add both to find 2Sn: n(2a + (n-1)d)
Divide by 2
Alternating sequences
Have a negative coming ratio
Un for a geometric sequence
a x r^(n-1)
How to find the common ratik
Divide two sets of consecutive numbers and set them equal to each other
Sum of geometric series if |r| <1
a(1-r^n) / 1-r
Sum of a geometric sequence if |r| > 1
a(r^n -1) / r-1
Prove the sum of a geometric series
Sn = a + ar + ar^2 … + ar^(n-1)
Multiply everything by r
Take away (2) from (1)
Sn - rSn = a - ar^n
When is sum to infinity value
If, for a geometric sequence, as n -> infinity, |r|<1
Convergent (not divergent)
Equation for sum to infinity
a / (1-r)
For recurring decimals
Set up a geometric series
If in doubt
Sub in a couple of values to establish series variables
If given a combination of arithmetic and geometric series
Split into 2
If asked to calculate Sk
K cannot be negative, and must be an integer, so round up
Test if something is geometric
By subbing in values and seeing if there is a common ratio
Types of function
- one-to-one
- many-to-one
MUST have distinct output
NOT one-to-many
If you have to draw a graph of range with discrete numbers
Do not join the dots
When do u need to give an explanation for an exclusion
ALL THE TIME
To sketch a graph from a parametric, set up a table of values
Use domain given for t and sub into parametrics
Domain of f(x)
Range of x(t)
Range of f(x)
Range of y(t)
Sketching parametrics tips
- must indicate direction of flow
* May be multiple revolutions within the domain
Finding x-intercept
- set y(t) = 0
- solve for t
- sub t into x(t)
To find E (a,b)
- set x(t) = à
* set y(t) = b
To find intersection
Solve for t by subbing parametrics into the Cartesian and solving simultaneously
Sub t back into parametrics
Showing a tangent
Only 1 intersection
Only 1 t value
Vertical velocity
Vsinθ
Horizontal velocity
Vcosθ
Vertical distance
(Vsinθ)t
Horizontal distance
(Vcosθ)t
Why are parametric models unrealistic
Values cannot continue tor use indefinitely
Particles projected from a height
- horizontal velocity is unaffected by gravity
- vertical velocity is only Vsinθ initially, after than u must use suvat
s = y(t) - height of projection a = -9.8 t = t
s = ut + 1/2at^2 y(t) = (Vsinθ)t -4.9t^2 + height of projection
Periodic graphs
Period gives the time taken to return to horizontal position
Unaffected by shifts
