What is the next term in this sequence: 2, 9, 28, 65, 126, 217, _____?
Identifying Patterns
Unusual Sequence
Here is a rather unusual sequence: 125, 59, 42, 34, 14, 4, ____. Can you finish the sequence?
Six Sequences to Continue
Continue these sequences. Explain how each one works.
| a. | 0 | 1 | 3 | 6 | 7 | 9 | 12 | 13 | ___ | ___ | ___ |
| b. | 1 | 2 | 3 | 4 | 9 | 8 | 27 | 16 | ___ | ___ | ___ |
| c. | 0 | 4 | 8 | 4 | 8 | 12 | 6 | 10 | ___ | ___ | ___ |
| d. | 1 | 2 | 3 | 3 | 6 | 9 | 9 | ___ | ___ | ___ | ___ |
| e. | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | ___ | ___ | ___ |
| f. | 0 | 1 | 1 | 2 | 4 | 7 | 13 | 24 | ___ | ___ | ___ |
Sit-Ups
To make the team, you are going to have to do 89 sit-ups for the coach a week from today. You decide to work up to it. You will start by doing 3 sit-ups today (no sense rushing into things) and end on the 8th day with 89. You don’t know how many you will do tomorrow, but you decide that from the 3rd day on, the number of sit-ups you do will be the sum of what you did on the two preceding days. That is, the number you do on Wednesday will be the sum of the number you did on Monday and the number you did on Tuesday; the number you do on Thursday will be the sum of what you did on Tuesday and Wednesday, and so on. Find out how many sit-ups you should do tomorrow to make this work, so that you come out with 89 a week from today.
Giant Binomial Squared
The sum of the digits in base ten of (10(4n2 +8) +1)2, where n is a positive integer, is:
- 4
- 4n
- 2 + 2n
- 4n2
- n2 + n + 2
Alternating Sums of Squares
Evaluate 12 – 22 + 32 – 42 + 52 – 62 + … + 1992.
Sequence Formula
Consider the numbers Fn defined by the following formula:
where n may be any positive integer. Calculate F1 through F5. Do you notice anything suprising?
Defined Operations 2; Do Composition
If 3 h = 10, 7 h = 50, 5 h = 26; and 4 b = 1, 7 b = 2.5, 20 b = 9, then what is n
if n hb = 17.5?
Defined Operations 3; Do Composition
If 10 g = 16, 100 g = 196, 4 g = 4 and 12 r = 5; 300 r = 101, 30 r = 11, then what is n if n g r = 9?
Function as Average
Let xk = (-1)k for any positive integer k. Let f(n) = (x1 + x2 + … + xn)/n, where n is a positive integer. Give the range of this function.
- 0
- 1/n (where n is any positive integer)
- 0 and -1/n (where n is any odd positive integer)
- 0 and 1/n (where n is any positive integer)
- 1 and 1/n (where n is any odd positive integer).