# Thread: number of terms in geometric progression

1. ## number of terms in geometric progression

obtain the number of terms in the following geometric progression

81, 27 , 9 , 1/243

can i find the number of terms using this

$\displaystyle ar^{n-1} = 1/243$

the answer that i get is a fraction though and that couldnt be possibly correct

2. I suppose there are some skipped terms between 9 and 1/243.

So $\displaystyle a = 81 = 3^4$ and $\displaystyle r = 1/3$. The equation is $\displaystyle 3^4(1/3)^{n-1}=1/243=3^{-5}$. From here, $\displaystyle (1/3)^{n-1}=3^{-9}$, or $\displaystyle 3^{1-n}=3^{-9}$, so n = 10.

3. Hello, sigma1!

$\displaystyle \text{Obtain the number of terms in the following geometric progression:}$

. . . . $\displaystyle 81,\;27,\;9,\;\hdots\;\dfrac{1}{243}$

This geometric progression has: .$\displaystyle \begin{Bmatrix}\text{first term:} & a_1 &=& 81 \\ \text{common ratio:} & r &=& \frac{1}{3} \end{Bmatrix}$

$\displaystyle \text{The }n^{th}\text{ term is: }\:a_n \:=\:a_1r^{n-1}$

So we have: .$\displaystyle a_n \;=\; 81\left(\dfrac{1}{3}\right)^{n-1} \;=\;3^4\cdot\dfrac{1}{3^{n-1}} \;=\;\dfrac{1}{3^{n-5}}$

We want: .$\displaystyle a_n \:=\:\dfrac{1}{243}$

. . $\displaystyle \dfrac{1}{3^{n-5}} \:=\:\dfrac{1}{3^5} \quad\Rightarrow\quad n - 5 \:=\:5 \quad\Rightarrow\quad n \:=\:10$

Therefore, there are ten terms in the progression.

4. Originally Posted by emakarov
I suppose there are some skipped terms between 9 and 1/243.

So $\displaystyle a = 81 = 3^4$ and $\displaystyle r = 1/3$. The equation is $\displaystyle 3^4(1/3)^{n-1}=1/243=3^{-5}$. From here, $\displaystyle (1/3)^{n-1}=3^{-9}$, or $\displaystyle 3^{1-n}=3^{-9}$, so n = 10.
thanks alot.. i see where i went wrong.. if you were suposed to find the sum of the series though would you use this formula
$\displaystyle S_n = a(1-r^n) /1-r$

5. Originally Posted by sigma1
if you were suposed to find the sum of the series though would you use this formula
$\displaystyle S_n = a(1-r^n) /(1-r)$
Yes.