1. How many jumps will it take for the frog to reach the second leaf?
1 metre
A frog wishes to jump from one leaf to another, a distance of 1 metre. It can jump only 50cm on the first jump, then half the previous distance on each following jump.
How many jumps will it take for the frog to reach the second leaf?

2. On successive jumps, the frog jumps…...
50cm, 25cm, 12½cm, 6¼cm……...
The distance the frog jumps will be the sum of these terms…...
Sn = 50cm + 25cm + 12½cm + 6¼cm + ……...

3. This is a geometric series with a= 50, r = ½ & n = n
Sn = 50cm + 25cm + 12½cm + 6¼cm + ……...
This is a geometric series with a= 50, r = ½ & n = n
So, using
After n jumps, the frog covers

4. But 2n is being divided into 1
As the frog continues to jump
n will get bigger and bigger.
This will make 2n an even larger number!
But 2n is being divided into 1
So, as n gets bigger, will be even smaller 1 2n 1
As n gets bigger, the value of ( ) is always less than…. 1 2n
So the distance the frog covers will
always be less than 100cm

5. Not all GEOMETRIC SERIES have limiting sums……….
In n continues indefinitely, then we can say…..
As n  ,Sn  100(1 - 0) = 100
The “limiting sum”, or “sum to infinity” of the series
50cm + 25cm + 12½cm + 6¼cm + ……… is
100
Not all GEOMETRIC SERIES have limiting sums……….

6. For any geometric series,
a, ar, ar2, ar3, …, arn-1 ..
In the formula for the sum of a geometric series,
As n increases, only rn will be affected
So the possibility of a limiting sum occurring will depend on the value of r.
Case 1.
When -1 < r < 1
i.e. r < 1
i.e. r is a proper fraction
In this case, as n  , rn  0
And there IS a limiting sum.
So the sum  =
Case 2.
When r = 1
This time the sequence becomes, a, a, a,…..
There is a NO limiting sum.
So, Sn = n  a

7. So there is a NO limiting sum.
For any geometric series,
a, ar, ar2, ar3, …, arn-1 ..
Case 3.
When r = -1
This time the sequence becomes, a, -a, a, -a,…
There is a NO limiting sum.
So, Sn = oscillates between a and 0
Case 4.
When r > 1
As n increases, so does rn
You use the formula
There is a NO limiting sum.
Case 5.
When r < -1
So there is a NO limiting sum.
You use the formula
and…...
As n increases, rn increases without limit IF n is even
and…...
As n increases, rn decreases without limit IF n is odd

8. So, a VERY IMPORTANT SUMMARY:
For any geometric series,
a, ar, ar2, ar3, …, arn-1 ..
A limiting sum exists ONLY if -1 < r < 1
When the limiting sum exists, it equals a
1 - r
i.e.
When -1 < r < 1
Set 9H has plenty of limiting sum situations!