Sequences and Series

Question 1

Prove sum of arithmetic series formula

Answer 1

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

Question 2

Alternating sequences

Answer 2

Have a negative coming ratio

Question 3

Un for a geometric sequence

Answer 3

a x r^(n-1)

Question 4

How to find the common ratik

Answer 4

Divide two sets of consecutive numbers and set them equal to each other

Question 5

Sum of geometric series if |r| <1

Answer 5

a(1-r^n) / 1-r

Question 6

Sum of a geometric sequence if |r| > 1

Answer 6

a(r^n -1) / r-1

Question 7

Prove the sum of a geometric series

Answer 7

Sn = a + ar + ar^2 … + ar^(n-1)

Multiply everything by r

Take away (2) from (1)

Sn - rSn = a - ar^n

Question 8

When is sum to infinity value

Answer 8

If, for a geometric sequence, as n -> infinity, |r|<1

Convergent (not divergent)

Question 9

Equation for sum to infinity

Answer 9

a / (1-r)

Question 10

For recurring decimals

Answer 10

Set up a geometric series

Question 11

If in doubt

Answer 11

Sub in a couple of values to establish series variables

Question 12

If given a combination of arithmetic and geometric series

Answer 12

Split into 2

Question 13

If asked to calculate Sk

Answer 13

K cannot be negative, and must be an integer, so round up

Question 14

Test if something is geometric

Answer 14

By subbing in values and seeing if there is a common ratio

Question 15

Types of function

Answer 15

• one-to-one
• many-to-one

MUST have distinct output

NOT one-to-many

Question 16

If you have to draw a graph of range with discrete numbers

Answer 16

Do not join the dots

Question 17

When do u need to give an explanation for an exclusion

Answer 17

ALL THE TIME

Question 18

To sketch a graph from a parametric, set up a table of values

Answer 18

Use domain given for t and sub into parametrics

Question 19

Domain of f(x)

Answer 19

Range of x(t)

Question 20

Range of f(x)

Answer 20

Range of y(t)

Question 21

Sketching parametrics tips

Answer 21

• must indicate direction of flow

* May be multiple revolutions within the domain

Question 22

Finding x-intercept

Answer 22

• set y(t) = 0
• solve for t
• sub t into x(t)

Question 23

To find E (a,b)

Answer 23

• set x(t) = à

* set y(t) = b

Question 24

To find intersection

Answer 24

Solve for t by subbing parametrics into the Cartesian and solving simultaneously

Sub t back into parametrics

Question 25

Showing a tangent

Answer 25

Only 1 intersection

Only 1 t value

Question 26

Vertical velocity

Answer 26

Vsinθ

Question 27

Horizontal velocity

Answer 27

Vcosθ

Question 28

Vertical distance

Answer 28

(Vsinθ)t

Question 29

Horizontal distance

Answer 29

(Vcosθ)t

Question 30

Why are parametric models unrealistic

Answer 30

Values cannot continue tor use indefinitely

Question 31

Particles projected from a height

Answer 31

• 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

Question 32

Periodic graphs

Answer 32

Period gives the time taken to return to horizontal position

Unaffected by shifts