A boxer runs up stairs as part of her training - Junior Cycle Mathematics - Question 14 - 2015
Question 14
A boxer runs up stairs as part of her training.
She can go up 1 step or 2 steps with each stride, as shown.
The boxer wants to count how many different ways she can... show full transcript
Worked Solution & Example Answer:A boxer runs up stairs as part of her training - Junior Cycle Mathematics - Question 14 - 2015
Step 1
Find the value of T_1 and T_2.
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Answer
To find the Taylor numbers T_1 and T_2:
For T_1:
The only way to reach the first step is by taking a single step. Therefore, T_1 = 1.
For T_2:
The possible ways to reach the second step are:
1 step + 1 step
2 steps at once
Therefore, T_2 = 2.
Step 2
List all the different ways that she can reach the 4th step; one way is already done for you.
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Different ways to reach the 4th step include:
1 + 1 + 1 + 1
1 + 1 + 2
1 + 2 + 1
2 + 2
Thus, T_4 = 5.
Step 3
List the different ways that she can reach the 5th step, if she starts by going up 1 step.
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If she starts by going up 1 step, the ways are:
1 + 1 + 1 + 1 + 1
1 + 1 + 1 + 2
1 + 1 + 2 + 1
1 + 2 + 1 + 1
2 + 1 + 1 + 1
This totals to 5 ways.
Step 4
List the different ways that she can reach the 5th step, if she starts by going up 2 steps.
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If she starts by going up 2 steps, the ways are:
2 + 1 + 1 + 1
2 + 1 + 2
2 + 2 + 1
This totals to 3 ways.
Step 5
Explain why T_{100} = T_{99} + T_{98}.
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To reach the 100th step, she can either:
Start by going up 1 step, which means she has to reach the 99th step afterwards (T_{99} ways).
Start by going up 2 steps, which means she then has to reach the 98th step (T_{98} ways).
Thus, the total number of ways to reach the 100th step is given by the sum of the ways for the previous two steps: T_{100} = T_{99} + T_{98}.
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