Some geometric series questions

By Rania Kashi (P3223) on Sunday, November 5, 2000 - 07:54 pm :

Please help me answer the following questions on series and sequences:

1) A ball is dropped from a height of 10m and bounces on a horizontal floor to a height of 8m. On each successive bounce the hirght reached is 0.8 times the height reached in the previous bounce. Find the total distance travelled by the ball before it comes to rest.

2) The first three terms of a geometric series are 4x, x+1, and x. Given that x is negative find the sum to infinity of the series.

3) A geometric series has 4th term 10 000 and 8th term 1. find the two series which satisfy these data and find the sum to infinity of each of these series.

Thanks in advance!!!

Rania

By Olof Sisask (P3033) on Sunday, November 5, 2000 - 09:17 pm :

1) Consider this:

The ball first drops 10m.
The ball then bounces up 10 x 0.8 = 8m.
The ball then drops 8m. It then bounces up 8 x 0.8 = 6.4m, then drops 6.4m, and so on and on, so you get this series:

10 + 8 + 8 + 6.4 + 6.4 + ... ad infinitum.

Let S = 8 + 6.4 + 5.12 + ...
Then D, the total distance travelled, equals

10 + 2 xS

since every term, bar 10, occurs twice.
S is a geometric series summed to infinity, with common ratio 0.8 and first term 8. You can then plug these values into the formula for the sum to infinity of a geometric series

(

S_{\infty} = a / (1 - r)

, where

a

is the first term and

r

is the common ratio), and thus work out the answer.
I have to go now unfortunately, but hope that helps!

Regards,
Olof.

By Steve Megson (Smm47) on Sunday, November 5, 2000 - 10:26 pm :

Since Olof has to go, I'll say something about 2 and 3.

2) In order to find the sum to infinity of the series we need to know the first term $a$ and the common ratio $r$ .

We have the three equations

(1) $a = 4 x$

(2) $a r = x + 1$

(3) $a r^{2} = x$

We can combine (1) and (2) to give

$(x + 1) = r (4 x) \Rightarrow r = (x + 1) / (4 x)$

Combining (2) and (3) gives $x = r (x + 1)$ and substituting for $r$ gives

$x = (x + 1)^{2} / (4 x)$

$\Rightarrow 3 x^{2} - 2 x - 1 = 0$

$\Rightarrow x = 1$ or $x = - 1 / 3$ , but we are told that $x$ is negative

Then $a = 4 x = - 4 / 3$ and $r = (x + 1) / 4 x = - 1 / 2$

You can now find $S_{\infty}$ using the equation Olof gave.

3) This is a very similar question, with the equations

$a r^{3} = 10000$

$a r^{7} = 1$

This gives $r^{4} = 1 / 10000 \Rightarrow r = \pm 1 / 10$

Then $a (\pm 1 / 107) = 1 \Rightarrow a = \pm 107$

This gives the two series, and you can find their sums.

If you need any more help, let me know.

By Rania Kashi (P3223) on Monday, November 6, 2000 - 12:11 am :

thanks a lot you guys, you are lifesavers! I can see what I had to do now and I actually get it!!! and now i can sleep peacefully. good night all.