Sequences/Series

Question 1

inductive series

Answer 1

each term is defined relating to its previous term

Question 2

Arithmetic sequence

Answer 2

A sequence when consecutive terms differ by the addition of a fixed number

Question 3

a

Answer 3

first term

Question 4

l

Answer 4

last term

Question 5

d

Answer 5

common difference

Question 6

n

Answer 6

number of terms

Question 7

Geometric sequence

Answer 7

A sequence where each term is found by multiplying the previous term by a fixed number

Question 8

r

Answer 8

common ratio

Question 9

geometric sequence example in letters

Answer 9

a, ar, ar^2, ar^3, ar^4…

Question 10

Periodic sequence

Answer 10

a sequence which repeats itself at regular intervalsp

Question 11

period

Answer 11

after how many terms does the sequence repeat

Question 12

increasing

Answer 12

each term in the sequence is greater than the previous term

Question 13

decreasing

Answer 13

each term in the sequence is less than the previous term

Question 14

diverging sequence

Answer 14

difference between terms gets greater towards a point

Question 15

converging

Answer 15

difference between terms gets less towards a point

Question 16

last term in an arithmetic sequence

Answer 16

l = a +(n-1)d

Question 17

Summation of an arithmetic sequence with last term

Answer 17

S = 1/2n(a+l)

Question 18

summation of an arithmetic sequence without last term

Answer 18

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

Question 19

Summation of terms in geometric sequence

Answer 19

S = (a(1-r^n))/(r-1)
OR
S = (a(1-r^n))/(1-r)

Question 20

Summing geometric sequences to infinity

Answer 20

for -1 0 as n->∞
therefore S = a/(1-r) as n increases
the series converges and has a sum to infinity

Question 21

examples of geometric growth

Answer 21

compound interest
population growth
radioactive decay

Question 22

sum of numbers where the first term is 1 and difference is 1 (sum of all natural numbers

Answer 22

1/2n(n+1)

Question 23

sum of the square of the first n natural numbers

Answer 23

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

Question 24

sum of the cubes of the first n natural numbers

Answer 24

1/4n^2(n+1)^2

Question 25

Method of differences

Answer 25

1. write the series as two or more series
2. sub in a few beginning term and a few ending terms
3. cancel out the middle terms appropriately

Question 26

proof by induction for um of series

Answer 26

1. prove its true for n=1
2. assume true for n=k
3. work out n=k+1 by
4. show this is equal to k+1 substituted for n
5. conclusion