Sequences and series

**ansonbound** · February 28th 2010, 07:26 AM

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.

**HallsofIvy** · February 28th 2010, 08:06 AM

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 $ar^{n-1}$ . Knowing that the second term is 24 tells you that $ar^{2-1}= ar= 24$ and knowing that the 5th term is 3 tells you that $ar^{5-1}= ar^4= 3$ . That gives you two equations to solve for a and r. In particular, if you divide the second equation by the first,
$\frac{ar^4}{ar}= r^3= \frac{3}{24}= \frac{1}{8}$

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

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

**masters** · February 28th 2010, 08:11 AM

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

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

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

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

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

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:

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

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

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

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

$S=96$

**ansonbound** · February 28th 2010, 02:39 PM

THANKS SO MUCH!!
But im stuck on this question now... only b)

http://www.mathhelpforum.com/math-he...rst-terms.html

**HallsofIvy** · March 1st 2010, 02:17 AM

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

Something is missing! How many terms?

**Bacterius** · March 1st 2010, 02:41 AM

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 ?

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