P2.3 Sequences & Series

Question 1

what is the difference b/w consecutive terms in arithmetic sequences?

Answer 1

constant = common difference
if common difference is +ve, the sequence is increasing
if common difference is -ve, the sequence is decreasing

Question 2

what is the formula for nth term of an arithmetic sequence?

Answer 2

Un = a + (n-1)d
a - first term
d - common difference

Question 3

what is an arithmetic series?

Answer 3

the sum of the terms of an arithmetic sequence

Question 4

what is the sum of the first n terms of an arithmetic series?

Answer 4

Sn = n/2 (2a + (n-1)d)

Sn = n/2 (a+l)

Question 5

what is the method to prove the sum of the first n terms of an arithmetic sequence?

Answer 5

1. write out the sum of the terms
   Sn = a + (a+d) + (a+2d)…(a+(n-2)d) + (a+(n-1)d)
2. write out the sum reversed
   Sn = (a+(n-1)d) + (a+(n-2)d) + (a+2d) + (a+d) + a
3. add together & divide by 2
   2Sn = n(2a + (n-1)d)
   Sn = n/2 (2a + (n-1)d)

see pg 63 P2 textbook

Question 6

what is a geometric sequence?

Answer 6

has a common ratio b/w consecutive terms
to get from one term to the next, multiply by the common ratio

Question 7

what is the formula for the nth term of a geometric sequence?

Answer 7

Un = ar^(n-1)
a is the first term
r is the common ratio
using logs to find n

Question 8

what is the formula for the sum of the first n terms of a geometric series?

Answer 8

Sn = a(1-r^n) / 1-r when r<1

Sn = a(r^n-1) / r-1 when r>1

Question 9

what is the proof for the formula for the sum of the first n terms of a geometric series?

Answer 9

see OneNote/p71 P2 textbook

Question 10

what happens when r is -ve?

Answer 10

it creates an alternating sequence

Question 11

what is true for in a convergent sequence?

Answer 11

|r| < 1
terms tend towards a limit

Question 12

what is true for a divergent sequence?

Answer 12

|r| > 1
S∞ for a divergent sequence is ∞

Question 13

what is the formula for the sum to infinity for a convergent series?

Answer 13

S∞ = a / 1-r

Question 14

prove the sum to infinity formula

Answer 14

see OneNote/pg 73 P2 textbook

Question 15

what does sigma signify

Answer 15

a sum
limits written on the top & bottom of the symbol Σ to show which terms you are summing

Question 16

describe how to solve problems involving Σ
& practice

Answer 16

sub in values of r to find the terms of the sequence - r does not have to start at one
see OneNote

Question 17

what is a recurrence relation?

Answer 17

the rule to get from one term to the next term
Un+1 = f(Un) defines each term of a sequence as a function of the previous term

Question 18

increasing vs decreasing sequences

Answer 18

increasing if Un+1 > Un
decreasing if Un+1 < Un

Question 19

describe periodic sequences

Answer 19

sequence with terms that repeat in a cycle
there is an integer k such that Un+k = Un
k is the order of the sequence

always write out the first few terms of the sequence

Question 20

describe modelling with series

Answer 20

geometric sequence: increasing/decreasing by the same percentage each year

geometric series: models the amount in total over n years

Question 21

modelling with series practice