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.

## Number Sense and Logical Reasoning

## 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 | ___ | ___ | ___ |

## Unusual Sequence

Here is a rather unusual sequence: 125, 59, 42, 34, 14, 4, ____. Can you finish the sequence?

## One Sequence to Continue

What is the next term in this sequence: 2, 9, 28, 65, 126, 217, _____?

## 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 *ax*^{3} + *bx*^{2} + *cx* = *d*

## Alternating Sums of Squares

Evaluate 1^{2} – 2^{2} + 3^{2} – 4^{2} + 5^{2} – 6^{2} + … + 199^{2}.

## Arithmetic Mean of Logs

## Giant Binomial Squared

The sum of the digits in base ten of (10^{(4n}^{2} +8) +1)^{2}, where *n* is a positive integer, is:

- 4
- 4
*n* - 2 + 2
*n* - 4
*n*^{2} *n*^{2}+*n*+ 2

## Postage Function, Using [ ]

If the postal rate for first class letters is 42 cents for the first ounce or any portion thereof, and 17 cents for every ounce or portion thereof after the first ounce, then find the cost of a letter weighing *w* ounces, where *w* is some positive real number. Use [*x*] to stand for the greatest integer in *w*, as usual.

## 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?