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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?. On successive jumps, the frog jumps….

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How many jumps will it take for the frog to reach the second leaf?

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

  4. So, as n gets bigger, will be even smaller 1 2n 1 As n gets bigger, the value of (1 - ) is always less than…. 2n 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 1 So the distance the frog covers will always be less than 100cm

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

  6. a, ar, ar2, ar3, …, arn-1 .. In this case, as n , rn  0 So the sum  = For any geometric series, 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. When -1 < r < 1 i.e. r < 1 i.e. ris a proper fraction Case 1. And there IS a limiting sum. When r = 1 This time the sequence becomes, a, a, a,….. Case 2. There is a NO limiting sum. So, Sn = n  a

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

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