Math 100. Exploring Mathematics. Spring 2008.
Explorations with patterns (continued)
1. Find the following sums:
What do you notice? Do you think the pattern will continue? Why? Calculate 1+3+5+7+...+97+99.
2. Calculate the following the differences:
- 1/1 - 1/2
- 1/2 - 1/3
- 1/3 - 1/4
- 1/4 - 1/5
What do you notice? What is the value of 1/99-1/100?
3. Calculate the following sums (reduce your answers):
- 1/(1*2)
- 1/(1*2)+1/(2*3)
- 1/(1*2)+1/(2*3)+1/(3*4)
- 1/(1*2)+1/(2*3)+1/(3*4)+1/(4*5)
- 1/(1*2)+1/(2*3)+1/(3*4)+1/(4*5)+1/(5*6)
What do you notice? Do you think the pattern will continue? Why? (Hint: use problem 2 to write each fraction as a difference, then cancel terms.)
What is the value of 1/(1*2)+1/(2*3)+1/(3*4)+...+1/(98*99)+1/(99*100)?
4. The sequence 1, 1, 2, 3, 5, 8, 13, ... , where each number is equal to the sum of two previous numbers, is called the
Fibonacci sequence. The numbers in this sequence are called Fibonacci numbers.
- Calculate the next 4 Fibonacci numbers.
- Determine the parity (i.e. whether it is even or odd) of each number in the sequence and describe the pattern. Do you think the pattern will continue? Explain why.
(That is, explain how you can be sure that it will.)
5. Calculate the following sums:
- 1
- 1+2
- 1+2+3
- 1+2+3+4
- 1+2+3+4+5
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- 13
- 13+23
- 13+23+33
- 13+23+33+43
- 13+23+33+43+53
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Compare the two sequences you get. What do you notice? Describe the pattern in words and write a formula. Can you explain why this pattern always holds, no matter how
far you go in the sums?
This page was last revised on 30 January 2008.
