Sequences and Series

Question 1

Arithmetic sum:

Answer 1

Sn = 0.5n(2a+(n-1)d)

= 0.5n(a+l)

Question 2

Geometric sum:

Answer 2

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

Question 3

Arithmetic term:

Answer 3

a + (n-1)d

Question 4

Geometric term:

Answer 4

ar^n-1

Question 5

Sum to infinity:

Answer 5

a/1-r

Question 6

Two series might be interleaved so

Answer 6

find the sum of each one and addd

Question 7

For oscillating series between positive and negative

Answer 7

find the running total

Question 8

A typical trick is to replace xn+1 with

Answer 8

the expressions involving xn and yn, then simplifying

Question 9

Multiple series interleaved

Answer 9

find sum of interleaved

Question 10

Oscillating positive and negative

Answer 10

Consider the running total

Question 11

(n+1)th term

Answer 11

replace n with n+1`

Question 12

A typical trick is to replace xn+1 with the expressions involving xn and yn

Answer 12

Sometimes you can reapply your recurrence relationship to obtain larger values needed

Question 13

Just writing xk+1 in two different ways here (in terms of both Ak and Ak+1 for example) and then comparing coefficients, will do the trick.

Answer 13

We’re comparing parameters from the current and next term in the sequence

Question 14

Recurrence relationships

Answer 14

compare coefficients

Question 15

If n>= a,

Answer 15

Sub in a to make series simpler, or the simplest number allowed

Question 16

look for multiple series’ in a single series

Answer 16

find sum of each

Question 17

Find patterns in function sequences,

Answer 17

whys this true

Question 18

When cannot find series formula

Answer 18

check that all iterations are correct

Question 19

recurrence:

Answer 19

relate xn+1 to xn, try values

Question 20

Try and relate separate recurrence formula to each other

Answer 20

series find patterns

Question 21

xn > xn+1 –>

Answer 21

n^2 + 2n > (n+1)^2 + 2(n+1)

Question 22

f(n) series… patterns

Answer 22

Recurrence formula: compare coefficients

Question 23

List results find pattern between n and f(n) but

Answer 23

check that all iterations are correct

Question 24

Oscillating series

Answer 24

split into two

Question 25

Finding a formula from recurrence relationship: list answers

Answer 25

Use recurrence relationships

Question 26

Base cases, n be as small as possible in order to simplify problem

Answer 26

x = a + (n-1)d

Question 27

∑n,r=1 (r) =

Answer 27

n(n+1)/2

Question 28

∑n,r=1 (r^2) =

Answer 28

(2n+1)(n+1)/6

Question 29

∑n,r=1 (r^3) =

Answer 29

0.25n^2(n+1)^2

Question 30

∑n,r=1 (k) =

Answer 30

kn

Question 31

∑n,r=1 (ar+b) =

Answer 31

bn + a∑n,r=1 (r)

Question 32

∑n,r=k (r) =

Answer 32

∑n,r=1 (r) - ∑k-1,r=1 (r)