sequences and series

Question 1

Answer 1

main point here : USE QUADRATIC FORMULA IF THEY ASK YOU TO DO IT IN THE SQUARE ROOT FORM !

Question 2

sum of arithmetic series

Answer 2

Question 3

PROOF of sum of arithmetic series (HAVE TO LEARN !)

Answer 3

Question 4

prove the the sum of 1-100 is 5050

Answer 4

Question 5

how would you find the number of terms in a sequence?

eg..

Answer 5

equate last value with a + (n-1)d formula !!!!!! and solve for “n”

eg …

Question 6

WHAT does n represent?

Answer 6

NUMBER of terms in a sequence!

Question 7

Answer 7

note; if theres not enough to fill the (k+1)th term, then sn is smaller than the number of samples given

Question 8

In “x, 3, x+6” geometric sequence… why cant x be the negative answer when solving ?

Answer 8

because there are no negative terms in the sequence therefore x cannot be negative (this is because it says in the original question that ALL terms are POSITIVE )_

Question 9

is “x” term in the sequence ? (1)

Answer 9

yes, as n is an interger

Question 10

Answer 10

remember: “n” has to be an interger therefore n= 12

Question 12

Answer 12

NOTE: rearrange for “a” MANUALLY !

( MISTAKE I DID HERE: i rearranged in my head and got it wrong , i didnt notice the “r^4” when multiplying bot sides by “(1-r)^2” therefore i got it wrong

Question 13

sum to infinity ; larger= divergent

smalle= convergent!

to show that a series is CONVERGENT, show that “r” is less than 1 !!!!!!!!!!!!

Answer 13

covergent = (modulus)”r”(modulus) is less than 1 therefore the values get smaller

Question 14

SUM TO INFINITY FORMULA

Answer 14

a/ 1-r

Question 15

note : take each value i nteh sequence as a seperate value in the sequence.

THIS IS ALSO AN EXAMPLE OF HOW YOU SHOUD ALWAYS MODFIY MODULUS SO THAT ITS PSOTIVE !

Answer 15

Question 16

Answer 16

NOTE; FOR “n” YOU HAVE TO INCLUDE THE FIRST VALUE TOO . so if the upper limit is 100 and the starting value is “7”, then the number of values in the sequence up to 100 is “94” NOT “93” BECAUSE WE HAVE TO INCLUDE THE “7”

steps to solving sigma notation problem

1. Get “a”

2. get the last value

3. get common ratio
4. sub into n/2(a+L)

Question 17

find this in terms of “k” ?

and how do we know “n”

Answer 17

“n”= k-9 since the starting value is “10”. therefore there are “k-9” values between 10 and “k”

another example: say k= 100

therefore there “100-9” values between 10 and 100 BECAUSE 10 IS ALSO INCLUDED

Question 18

recurrence relation question ( do part b)

Answer 18

simply do the sequence as normal and add all of them up and THEN SOLVE

Question 19

if a sequence is increasing

if its decreasing

a sequence is periodic if

Answer 19

un +1 is larger than un

un + 1 is smaller than un

it repeats in a cycle, state the roder of the periodic sequence (the order is the number of terms in one period, which is repeated in the periods that follow it )

Question 20

how would we go about solvign a periodic sequence

Answer 20

since its periodic, equate the repeating terms and find what yo uwant

( note, first write out the sequences innit bruv)

Question 22

suggest why salary increase may be unsuitable model

Answer 22

salary increase owuldnt be consistent every year

Question 24

what did i do wrong here?

Answer 24

i let the minimum speed be “ar” instead of “a”

Question 25

note: you can write “years” like this in sequences and series question….

Answer 25