Question

find r and a if S_5 = 61 and a_5=81

Answer 1

what does a_5 mean? and where's r in the question?

Answer 2

this a geometric sequence topic :)

Answer 3

ok so \(S_5\) is the sum right?

Answer 4

yup

Answer 5

\[S_5=a+ar+ar^2+ar^3+ar^4\] it is the sum of the first five terms

Answer 6

S_5=(A_1*rA_n)/(1-r)

Answer 7

do you know the formula for adding this up?

Answer 8

not quite

Answer 9

S_5=A_5*r^4 ?

Answer 10

you can use
\[\frac{a_1(1-r^n)}{1-r}\]

Answer 11

so
\[81=\frac{a_1-a_1r^4}{1-r}\] and also \(a_1r^4=61\) so we have
\[81=\frac{a_1-61}{1-r}\] hmmm

Answer 12

we use another equation

Answer 13

oh what equation?

Answer 14

A_5=A_1*r^4 ?
so that
81=A_1*r^4
A_1=81/r^4
then substitute?

Answer 15

hold on let me write it, but are you sure it is \(S_5=61\) and not \(S_5=60\)?

Answer 16

yes po :)

Answer 17

too bad because if it was 60 it would be easy, \(a_1=1, r=-3\)

Answer 18

uhmm...can we use another eq. for S_5? :)

Answer 19

lets try
\[\frac{a_1-a_1r^4}{1-r}\] and plug in what we know
\[\frac{a_1-81}{1-r}=60\]

Answer 20

cause honestly i already solve this and i got r=1 I just dont know if thats correct just wanna check

Answer 21

it cannot be right, if r = 1 it is not geometric, it is constant

Answer 22

also the denominator would be undefined

Answer 23

ohhkay.. :)

Answer 24

:(

Answer 25

my problem is that we have one equation and two variables, but i have yet to use the fact that it is geometric
what i am really thinking is that there may be infinite solutions

Answer 26

lets say we do as you suggested and write \(a_1=\frac{81}{r^4}\)

Answer 27

then we have
\[\frac{\frac{81}{r^4}-81}{1-r}=60\] oh maybe we can solve this for \(r\)

Answer 28

61 hehe

Answer 29

damn

Answer 30

itll be 20r^5 + 61r^4 - 81= 0

then we use quadratic form. to find r?

Answer 31

maybe my algebra is bad but that is not what i got

Answer 32

wat u got?

Answer 33

let me write it out

Answer 34

\[\frac{\frac{81}{r^4}-81}{1-r}=61\]
\[\frac{81}{r^4}-81=61-61r\]
\[\frac{81}{r^4}+61r-142=0\] or
\[61r^5-142r^4+81=0\]

Answer 35

being a fifth degree polynomial means there is no way to solve it unless you get lucky and guess

Answer 36

http://www.wolframalpha.com/input/?i=%28%2881%2Fx^4%29-81%29%2F%281-x%29%3D61

Answer 37

wat formula did u use for S_5?

Answer 38

\(S_5=\frac{a_1(1-r^4)}{1-r}\)

Answer 39

is that not what you are using?

Answer 40

r^4

Answer 41

?

Answer 42

oh crap
yeah \(r^5\)!!

Answer 43

yeah i am stuck
sorry
81 looks suspiciously like \((-3)^4\) and if you use \(=1,r=-3\) unfortunately you get a sum of 61 and not 60
i have no idea how you are supposed to solve a polynomial of this degree, although there was a solution in the worlfram link i sent

Answer 44

lololol you have 61!!!!

Answer 45

i need coffee
ok solution is done
\[a=1,r=-3\] that is all and it is an easy check
\[1-3+9-27+81=61\] finished!!!

Answer 46

lol okay thanks i forgot we can always make our own solution :) thanks thanks :)