1. ## Sequences and series

The second term of a geometric series is 24 and the fifth term is 3.
a. Show that the common ratio of the series is 1/2.
b. Find the first term of the series.
c. Find the sum to infinity of the series.

2. Originally Posted by ansonbound
The second term of a geometric series is 24 and the fifth term is 3.
a. Show that the common ratio of the series is 1/2.
b. Find the first term of the series.
c. Find the sum to infinity of the series.
That's pretty straightforward- if you know the definitions. The "nth" term of a geometric series with first term a and common ratio r is $\displaystyle ar^{n-1}$. Knowing that the second term is 24 tells you that $\displaystyle ar^{2-1}= ar= 24$ and knowing that the 5th term is 3 tells you that $\displaystyle ar^{5-1}= ar^4= 3$. That gives you two equations to solve for a and r. In particular, if you divide the second equation by the first,
$\displaystyle \frac{ar^4}{ar}= r^3= \frac{3}{24}= \frac{1}{8}$

Once you know r, it is easy to use $\displaystyle ar= 24$ to find a.

And, once you know both a and r, the "sum to infinity" is just
$\displaystyle \frac{a}{1- r}$

3. Originally Posted by ansonbound
The second term of a geometric series is 24 and the fifth term is 3.
a. Show that the common ratio of the series is 1/2.
b. Find the first term of the series.
c. Find the sum to infinity of the series.
Hi ansonbound,

____, 24, ____, ____, 3

Using $\displaystyle a_n=a_1 \cdot r^{n-1}$, we can verify that the common ratio is 1/2

$\displaystyle a_n=3$
$\displaystyle a_1=24$
$\displaystyle n=4$

$\displaystyle 3=24 \cdot r^{4-1}$

$\displaystyle \frac{1}{8}=r^3$

$\displaystyle r=\frac{1}{2}$

Working to the left of 24, the first term if found by multiplying 24 by 2. Thus, the first term is 48.

The sum S of an infinite geometric series with -1 < r < 1 is given by:

$\displaystyle S=\frac{a_1}{1-r}$

Since $\displaystyle r=\frac{1}{2}$, and since $\displaystyle \left|\frac{1}{2}\right| <1$, the sum exists.

$\displaystyle S=\frac{48}{1-\frac{1}{2}}$

$\displaystyle S=\frac{48}{\frac{1}{2}}$

$\displaystyle S=96$

4. THANKS SO MUCH!!
But im stuck on this question now... only b)
http://www.mathhelpforum.com/math-he...rst-terms.html

5. "Find the sum of the first ?? terms of the series."

Something is missing! How many terms?

6. Originally Posted by HallsofIvy
"Find the sum of the first ?? terms of the series."

Something is missing! How many terms?
Maybe you have to find out as well ?
I believe this question intends to make the student apply the formula on the first few terms. But it is indeed ambigious ... how many terms ?