Math: Sequences and Series

Question 1

formula for all odd numbers

Answer 1

a(n)=2n-1

a(n)= 2n+3 –> will always be odd, as well, since anything multiplied by 2 is even

Question 2

formula for perfect squares

Answer 2

a(n)= n^2

Question 3

formula for multiple of a number

Answer 3

a(n) = xn –> x being the number you’re finding multiples for

Question 4

(odd) (odd)

Answer 4

= odd

Question 5

arithmetic sequence formula

Answer 5

a(n) = a(1) + d(n-1)

d= common difference, or difference between two adjacent terms when you subtract or add them

Question 6

Finding arithmetic sequences through division and remainder

Answer 6

Another way to find an arithmetic sequence is if they give you a number to divide by and the remainder.

Ex. Let S be the positive set that when divided by 8, the remainder is 5. Find the 76th term.

First number is 5 –> 5 / 8 = 0 r 5
Difference is 8 (# that it’s divided by)

Question 7

Finding arithmetic sequence when you have formula, and two random numbers

Answer 7

two ways to solve this
1) subtract a numbers from each other and their respective totals, and then divide by one another to get the difference. You can then plug difference in to equation for either formula and get starting #

2) set-up the equations in their fullest form, subtract from one another and then solve for difference and then starting number

ex. a(3) = 17 and a(19)= 65
17= a(1)+2D and 65= a(1) + 18d – if you subtract eq. you get 48=16d —> d =3, plug back in to get difference

Question 8

recursive sequence

Answer 8

each term is defined by the previous term
noted by a(n-1)
since you need the prior term, it likely won’t ask you for the 40th term! you need to go one by one to get the value of the desired term
the numerical values of one or two terms will always be specified

Question 9

Fibonacci recursive sequences

Answer 9

Each term is the sum of the previous two terms

a(n)= a(n-1)+ a(n-2), for n>2

Question 10

inclusive counting

Answer 10

ordinary subtraction excludes the lowest end point.
you use inclusive counting, which is to add +1, to any ordinary subtraction when you want to ensure that both the starting # and the ending # (i.e. the endpoints, or the lowest value and the highest value) are included
ex. when counting number of days of a workshop - workshop started on the 8th and ended on the 27th – 27-8 = 19 +1 = workshop was held for 20 days.

Question 11

Inclusive counting: #s of multiples

Answer 11

When you need to find the # of multiples between two numbers, INCLUSIVE, you must add one to the total you find to include both endpoints.

ex. # of multiples of 8 between 200 and 640, INCLUSIVE
200/8 = 25
640/8 = 80
# of multiples = 80-25 +1 --> 56

Question 12

sum of sequences (generic)

Answer 12

n(n+1) / 2

Question 13

sum of arithmetic sequences

Answer 13

n*{(a(1)+a(n)] / 2
–> you could divide n/2 first - since it’s the # of pairs and that’s really what you’re counting in these sequences

ex. what’s the sum of the multiples of 20 from 160 to 840 inclusive.
First, found out how many multiples there are: 160/20 = 8 and 840/20=42 –> 42-8+1= 35 (inclusive)
35/2 = 17.5 (#of pairs) –> so then you can multiply by a(1) + a(n) –> 160+84017.5
alternatively you could do 160+84035 / 2