Home
Syllabus
Schedule
Screen Copy
Grader
Pre-Test
Billiards
Induction
Deduction
3 Ladies
3 Prisoners
Arithmetic
Pyramid
Geometric
Keyboard
Binary
8-Bit Adder
Squares
Cubes
Ring Game
Fibonacci
Phyllotaxis
Nim
Staircase
Counting
Flowers
Permutations
Duplications
Coin Flip
Combinations
Pascal's Tree
Texas Poker
Dice Rolls
Candychines
Lottery
Binomial ESP
More Dice
Monty Hall
Birthdays
BlackJack
Slot Machines
Ciphers
Today's Quote
Bell Curve
M&M Sampling
Worm Holes
Doodles
World Tour
CSG
Polys
Fractals
Chaos Game
Eggbrot

THE SEQUENCE OF CUBES In addition to picturing the square numbers as square arrays of dots, the ancient Greeks pictured other numbers with cubic arrays of dots. The number 8, for example, was pictured as a cube having 2 dots on each edge. Eight can also be represented as 2 x 2 x 2 or 23, which is read as “two cubed.” Cubes having 3 and 4 dots on each edge contain 3 x 3 x 3 = 33 = 27 dots and 4 x 4 x 4 = 43 = 64 dots, respectively. For this reason, 27 and 64 are called cube numbers. The sequence of cubes is       13 23 33 43 53 ... or, equivalently,       1 8 27 64 125 ... In the same way that square numbers are related to the area of a square (that is, the amount of surface that it occupies), cube numbers are related to the volume of a cube (the amount of three-dimensional space that it occupies). A square measuring 4 centimeters along each side has an area of 42 = 16 square centimeters; a cube measuring 4 centimeters along each edge has a volume of 43 = 64 cubic centimeters. Because the sequence of squares can be written as       12 22 32 42 52 ... and the sequence of cubes as       13 23 33 43 53 ... They are also referred to as the sequences of second and third powers, respectively. We have seen that second powers can be pictured in two dimensions using squares and third powers in three dimensions using cubes. Although we cannot picture sequences of powers higher than the third with arrays of dots (since this requires more than three dimensions), such sequences are both interesting and useful. The sequence of fourth powers,       14 24 34 44 54 or       1 16 81 256 625 can be applied, for example, to relating the luminosity of a star to its temperature. PENNIES & HEXAGONS These figures illustrate pennies arranged in the form of six-sided figures called hexagons. Use your imagination to “see” the hexagons. (The single penny will probably require the most imagination.) Fill in the table below. Then answer the questions. Copy and complete the following sequence of the numbers of pennies in each figure.       1 7 ? ? ? There is 1 penny in the first figure and there are 1 + 7 = 8 pennies altogether in the first two figures. How many pennies are there altogether in the first three figures? in the first four figures? in the first five figures? What do you notice about these numbers? VOLUMES & SURFACES The figures below illustrate cubes having edges of lengths 1, 2, 3, 4, and 5. The surface area of a cube is the number of square units on its six faces. The surface area of the first cube is       6 x 1 = 6 and the surface area of the second cube is       6 x 4 = 24 Copy and complete the following number sequence of the surface areas of the cubes shown.       6 24 ? ? ? The volume of a cube is the number of cubic units that it contains. Copy and complete the following number sequence of the volumes of the cubes shown.       1 8 ? ? ? The following table represents a sequence of cubes in which the length of the edge is successively doubled. Copy and complete the table. What happens to the surface area of a cube if the length of its edge is doubled? What happens to the volume of a cube if the length of its edge is doubled? SUM OF ODDS The following exercises refer to the pattern below. 1 = 1 3 + 5 = ? 7 + 9 + 11 = ? 13 + 15 + 17 + 19 = ? Copy it, filling in the missing numbers. What sequence do the numbers on the left sides of the equations form? What sequence do the numbers on the right sides of the equations form? Write the next two lines of the pattern. Are they also true? SUM OF CUBES The following exercises refer to this pattern. 1 = 1 1 + 8 = ? 1 + 8 + 27 = ? 1 + 8 + 27 + 64 = ? Copy it, filling in the missing numbers. To what sequence do the numbers on the left sides of the equations belong? To what sequence do the numbers on the right sides of the equations belong? Complete the table above to rewrite the pattern above using exponents. What does each row add up to be? Enter the totals in the Sum of Cubes column. What is a shortcut for finding the sum of the cubes on each line? Hint: find the numbers in the Square Root column. What is the next line of the pattern when the number of cubes is 7? Is it also true? SUM OF TRIANGULAR SQUARES Fill in the table below. Evaluate the expression n2(n+1)2/4 in column 2 for the n on its corresponding row. Find the consecutive differences between each row in column 3. What does the last number in column 2 on the 10th row represent (in 10 words or less)? If tn = 13 + 23 + 33 + ... + n3, what do you add to tn to find tn+1? tn+1 = 13 + 23 + 33 + ... + n3 + ? Write your answer as an expression of n. If tn = ¼n2(n+1)2 for any number n, then what is tn+1 ? Write your answer as an expression of n. Add (n+1)3 to ¼n2(n+1)2. Does (n+1)3 + ¼n2(n+1)2 = ¼(n+1)2(n+2)2? Prove it? In 10 words or less, what did you prove in the previous exercise? PACKING CUBES How many little cubes does the big cube contain? How many are 1x1x1 cubes? How many are 2x2x2 cubes? How many are 3x3x3 cubes? How many are 4x4x4 cubes? How many are 5x5x5 cubes? How many are 6x6x6 cubes? How many are 7x7x7 cubes? How many are 8x8x8 cubes? How many are 9x9x9 cubes? How many are 10x10x10 cubes? What is the sum of all the cubes? Add the answers from the previous 10 question. Fill in the table below starting with the number of Cubes in column 2. Add the cubes starting from the BOTTOM row upwards. Enter these partial sums into column 3 under Sums. Enter the Square Roots of the sums in column 3 into column 4. What is the name of this sequence of square roots?

Comments:

From wHolt - 9/22/08 6:45 PM

From wHolt - 9/22/08 4:48 PM