## 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:

1. 4
2. 4n
3. 2 + 2n
4. 4n2
5. n2 + n + 2

## Cubic to Quadratic to Linear

One root of a certain third-degree equation is 1. When the cubic term of the equation is crossed off, the resulting quadratic equation has a root of 2. When the squared term is also crossed off, the resulting linear equation has a root of 3. Reconstruct the original third-degree equation, expressing it in the form ax3 + bx2 + cx = d, with all coefficients as integers.

## 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.

1. 0
2. 1/n (where n is any positive integer)
3. 0 and -1/n (where n is any odd positive integer)
4. 0 and 1/n (where n is any positive integer)
5. 1 and 1/n (where n is any odd positive integer).