Geometric Sequences

[HNO]

1. Determine the missing terms t2, t3, and t4

t1= 8.1
t5= 240.1

I am trying to solve this but something is not working for me :/

tn(tn-1)r TO TRY AND FIND r

240.1 = (240.1-1)r

239.1= 240.1= 1.004 >>> that does not work

so then I tried


Tn = t1 r^n-1 ***** (n-1)= exponent

240 = (8.1) r^4 ---------29.6 x (1/4) = 7.4 but this is wrong as well

I was tolled you had to multiply 1/4 to get rid of the exponent 4

PLEASE HELP!

 

[soroban]

Hello, HNO!

\(\displaystyle \text{Geometric sequence with: }\:t_1 = 8.1,\;t_5 = 240.1\)

\(\displaystyle \text{Determine the missing terms: }\,t_2,\, t_3,\,t_4.\)

\(\displaystyle \text{We are given: }\:\begin{Bmatrix}t_1 &=& a &=& 8.1 & [1] \\ t_5 &=& ar^4 &=& 240.1 & [2]\end{Bmatrix}\)

\(\displaystyle \text{Divide [2] by [1]: }\:\dfrac{ar^4}{a} \:=\:\dfrac{240.1}{8.1} \:=\:\dfrac{2401}{81} \:=\:\dfrac{7^4}{3^4} \)

\(\displaystyle \text{Hence: }\:r^4 \:=\:\left(\frac{7}{3}\right)^4 \quad\Rightarrow\quad r \:=\:\pm \frac{7}{3}\)


\(\displaystyle \text{Therefore: }\:\begin{Bmatrix}t_2 &=& ar &=& 8.1\left(\pm\frac{7}{3}\right) &=& \pm18.9 \\ t_3 &=& ar^2 &=& 8.1\left(\pm\frac{7}{3}\right)^2 &=& 44.1 \\ t_4 &=& ar^3 &=& 8.1\left(\pm\frac{7}{3}\right)^3 &=& \pm102.9\end{Bmatrix}\)

 

[HNO]

Hello, HNO!


\(\displaystyle \text{We are given: }\:\begin{Bmatrix}t_1 &=& a &=& 8.1 & [1] \\ t_5 &=& ar^4 &=& 240.1 & [2]\end{Bmatrix}\)

\(\displaystyle \text{Divide [2] by [1]: }\:\dfrac{ar^4}{a} \:=\:\dfrac{240.1}{8.1} \:=\:\dfrac{2401}{81} \:=\:\dfrac{7^4}{3^4} \)

\(\displaystyle \text{Hence: }\:r^4 \:=\:\left(\frac{7}{3}\right)^4 \quad\Rightarrow\quad r \:=\:\pm \frac{7}{3}\)


\(\displaystyle \text{Therefore: }\:\begin{Bmatrix}t_2 &=& ar &=& 8.1\left(\pm\frac{7}{3}\right) &=& \pm18.9 \\ t_3 &=& ar^2 &=& 8.1\left(\pm\frac{7}{3}\right)^2 &=& 44.1 \\ t_4 &=& ar^3 &=& 8.1\left(\pm\frac{7}{3}\right)^3 &=& \pm102.9\end{Bmatrix}\)

HI!! THis stuf looks great but im not familiar with with the ar … I have never used a?

 

[stapel]

HI!! THis stuf looks great but im not familiar with with the ar … I have never used a?

The "a" is what you're calling "T1". The "r" is the same "r" you're using. :wink:

 

[HallsofIvy]

1. Determine the missing terms t2, t3, and t4

t1= 8.1
t5= 240.1

I am trying to solve this but something is not working for me :/

tn(tn-1)r TO TRY AND FIND r

There are, unfortunately, many people who think "learning mathematics" means "memorizing formulas". Your problem is that you have not even memorized the formula correctly.

First you left out the "=" here but that is clearly just a typo since you have it below. More importantly, the "tn-1" does NOT mean to subtract 1 from the term- it means subtract 1 from the index: \(\displaystyle t_n= (t_{n-1})r\). That is, each term is the previous term times r.
\(\displaystyle t_2= t_1r= 8.1r\).
\(\displaystyle t_3= t_2r= (8.1r)r= 8.1r^2\)
\(\displaystyle t_4= t_3r= (8.1r^2)r= 8.1r^3\)
\(\displaystyle t_5= t_4r= (8.1r^3)r= 8.1r^4= 240.1\)

Can you solve \(\displaystyle 8.1r^4= 240.1\)?

240.1 = (240.1-1)r

This is, as I said before, simply using a wrong formula.

239.1= 240.1= 1.004 >>> that does not work

so then I tried


Tn = t1 r^n-1 ***** (n-1)= exponent

240 = (8.1) r^4 ---------29.6 x (1/4) = 7.4 but this is wrong as well

I was tolled you had to multiply 1/4 to get rid of the exponent 4

No, you were never told that!

PLEASE HELP!