Year 2 chapter 3: Sequences And Series

Question 1

What is the key feature of an arithmetic sequence?

Answer 1

The difference between consecutive terms is constant.

Question 2

Give formula for the nth term of arithmetic sequence and give notation:

Answer 2

. Un = a +(n-1)d
. ‘a’ is first term of sequence.
. ‘d’ is common difference between consecutive terms.
. ‘n’ the number of the position of the term (1st,2nd,3rd,4th etc…)
. ‘Un’ is the nth term of the sequence.

Question 3

What does u3 = 5 mean?

Answer 3

The third term of the sequence is 5.

Question 4

What is an arithmetic series?

Answer 4

The sum of the terms of an arithmetic sequence.

Question 5

What is Sn equivalent to?

Answer 5

The sum of the first n terms of a series.

Question 6

Give 2 formulas for the sum of the first n terms of an arithmetic series (give notation for what ‘L’ is):

Answer 6

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

. Sn = n/2 (a + L)

Question 7

What is a geometric sequence?

Answer 7

. A sequence that has a common ratio (that isn’t 1) between consecutive terms.
. To get from one term to the next term, you multiply that one term by common ratio.

Question 8

What is the difference between a divergent and convergent sequence?

Answer 8

. A convergent sequence has common ratio of |r| <1, therefore the sequence tends to a certain value (this value is called the limit of the sequence).
. A divergent sequence has common ratio of |r|>1, therefore the sequence tends towards infinity (so infinity is the limit).

Question 9

Give the formula for the sum of the first n terms of a geometric series when |r|<1 and another formula for when |r|>1 (say what ‘r’ is):

Answer 9

• . |r|<1: Sn = a(1-r^n) / 1-r.
• . |r|>1: Sn = a((r^n)-1) / r-1.
• . ‘r’ is the common ratio of the sequence.

Question 10

Give a shortened formula for the sum of the first n terms of a convergent geometric series and the reason why this formula works:

Answer 10

• . Sn = a/(1-r).

- . This is because r^n tends towards 0 and n tends towards ♾.

Question 11

What does lim (n (arrow) ♾) mean?

Answer 11

• . The limit as n tends to ♾.
• . You can’t evaluate the expression when n is ♾ but as n gets larger the expression gets closer to a fixed (or limiting value).

Question 12

Give the formula for the sum to infinity of a convergent geometric series:

Answer 12

• S(♾) = a/(1-r)

Question 13

Give the formula for the nth term of a geometric sequence:

Answer 13

Un = ar^(n-1)

Question 14

What is sigma notation (briefly)?

Answer 14

. Involves the Greek capital letter ‘sigma’, which is used to signify a sum and is written as a weird shaped ‘E’.
. You write the limits on the top and bottom to show which terms you are summing.
. Used for arithmetic and geometric series.

Question 15

What does a sigma notation sum look like (with example and notation)?

Answer 15

. LHS: a capital ‘E’ (the Greek letter sigma) with ‘r=x’ on bottom of ‘E’ and ‘y’ on top of ‘E’ (where x and y are numbers).
- Then there is an algebraic expression in brackets with ‘r’ in that expression to the right of the ‘E’.
. RHS: a+b+c+d…..+p (where all these letters are the sum/series that the LHS gives).
. Example: E (r=1 on bottom and 5 on top) (2r-3) = -1+1+3+5+7
- ‘R=1’ means 1 is the lowest number that is replaces r in ‘2r-3’ and ‘5’ (from top of ‘E’) is the highest.
- R therefore is equivalent to what ‘n’ is in a normal arithmetic/geometric sequence.
- All the numbers between 1 and 5 are also subbed into ‘2r-3’ to make ‘1+3+5’.

Question 16

What are recurrence relationships and how is one written (give example and sequence for example)?

Answer 16

. It is a function that is written if you know the rule to get from one term to the next.
. In order to generate a sequence from a recurrence relation, you need to know the first term of the sequence (what u(1) is)
. Written as ‘u(n+1) = f(u(n))’.
. Example: u(n+1) = 2u(n) + 3, u(1)=6.
- sequence: 6, 15, 33, 69…
- the 15 comes from ‘2(6)+3’ then the 33 from ‘2(15)+3’.

Question 17

How do you know if the sequence of a recurrence relation is increasing or decreasing?

Answer 17

. Increasing if u(n+1) > u(n).

. Decreasing if u(n+1) < u(n)

Question 18

What is a periodic sequence (say what is looks like in terms of a recurrence relation) and how do you find the order of a periodic sequence?

Answer 18

. It is a sequence where terms repeat in a cycle.
. For a periodic sequence, there is an integer ‘k’ such that ‘u(n) + k = u(n)’.
- ‘n’ IS FOR ALL NATURAL NUMBERS, NOT JUST ONE CHOSEN NUMBER.
. The value ‘k’ is called the order of the sequence.