sequences and series Flashcards
main point here : USE QUADRATIC FORMULA IF THEY ASK YOU TO DO IT IN THE SQUARE ROOT FORM !
sum of arithmetic series
PROOF of sum of arithmetic series (HAVE TO LEARN !)
prove the the sum of 1-100 is 5050
how would you find the number of terms in a sequence?
eg..
equate last value with a + (n-1)d formula !!!!!! and solve for “n”
eg …
WHAT does n represent?
NUMBER of terms in a sequence!
note; if theres not enough to fill the (k+1)th term, then sn is smaller than the number of samples given
In “x, 3, x+6” geometric sequence… why cant x be the negative answer when solving ?
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 )_
is “x” term in the sequence ? (1)
yes, as n is an interger
remember: “n” has to be an interger therefore n= 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
sum to infinity ; larger= divergent
smalle= convergent!
to show that a series is CONVERGENT, show that “r” is less than 1 !!!!!!!!!!!!
covergent = (modulus)”r”(modulus) is less than 1 therefore the values get smaller
SUM TO INFINITY FORMULA
a/ 1-r
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 !
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
- get common ratio
- sub into n/2(a+L)
find this in terms of “k” ?
and how do we know “n”
“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
recurrence relation question ( do part b)
simply do the sequence as normal and add all of them up and THEN SOLVE
if a sequence is increasing
if its decreasing
a sequence is periodic if
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 )
how would we go about solvign a periodic sequence
since its periodic, equate the repeating terms and find what yo uwant
( note, first write out the sequences innit bruv)
suggest why salary increase may be unsuitable model
salary increase owuldnt be consistent every year
what did i do wrong here?
i let the minimum speed be “ar” instead of “a”
note: you can write “years” like this in sequences and series question….
