anonymous
  • anonymous
Find an equation for the nth term of a geometric sequence where the second and fifth terms are -21 and 567, respectively.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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ybarrap
  • ybarrap
A geometric sequence is a number that is related to the previous number by a certain factor, say \(r\): $$ a_n=r\times a_{n-1}=ra_{n-1} $$ We know that \(a_2=-21\) and \(a_{5}=567\) $$ a_2=ra_1=-21\\ a_3=ra_2=r(-21)\\ a_4=ra_3=r\times r(-21)=r^2(-21)\\ a_5=ra_4=r\times r^2(-21)=r^3(-21)=567\\ \implies r=\sqrt[1/3]{\cfrac{567}{-21}}=3 $$ This means that \(a_n=3a_{n-1}\) Make sense?
ybarrap
  • ybarrap
*correction $$ r=-3\\ a_n=-3a_{n-1} $$
anonymous
  • anonymous
7 • (-3)^n - 1 is that the answer

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DanJS
  • DanJS
the difference now, is instead of adding a number each time, now you are multiplying by a number each time
DanJS
  • DanJS
the number b, is what you multiply by each time moving up consecutive terms they gave you the second and fifth terms, they arent consecutive...
DanJS
  • DanJS
an = a0*(b)^n need to figure out a0 , and b (2 variables) they gave you two data points (n,an) = (2 , -21) (n , an) = (5 , 567)
DanJS
  • DanJS
those will give you two equations, you can solve those for b and a0
DanJS
  • DanJS
-21 = a0*(b)^2 and 567 = a0*(b)^5
DanJS
  • DanJS
or another way if you dont want to solve that system, moving from the second term to the 5th term, you are multiplying by 'b' 3 times -21*b^3 = 567 b = -3, right off the bat an = a0*(-3)^n
DanJS
  • DanJS
you can use any point, either of those two equations to get a0
anonymous
  • anonymous
thank you but i just want to know if my answer is right
anonymous
  • anonymous
7 • (-3)^n - 1
DanJS
  • DanJS
oh, you already know all this
anonymous
  • anonymous
did i get it right
DanJS
  • DanJS
what is that -1 at the end for
anonymous
  • anonymous
\[7 * (-3)^{n-1}\]
DanJS
  • DanJS
oh yeah, sorry, i have been missing that the entire time
anonymous
  • anonymous
thank you!!
DanJS
  • DanJS
a0 -21 = ao*-3^(2-1) a0 = 7 yeah
DanJS
  • DanJS
if it is too difficult to figure r, or can't, you can solve a system like above, except the pwers should be 1 less on those (n-1)

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