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THE SEQUENCE OF SQUARES In 1600 and something, Galileo discovered the law of gravity that all physics is based upon. Rolling a ball down a ramp, Galileo found how far a ball falls each second. TIME DISTANCE ACCELERATION  Seconds Meters  Meters/Sec/Sec   1  1  1  2  4  3  3  9  5  4  16  7  5  25  9  6  36  16 These numbers come from multiplying each counting number by itself:       1x1=1, 2x2=4, 3x3=9, 4x4=16, 5x5=25, ... Multiplying a number by itself is called squaring the number and is represented by writing a small 2 at the upper right of the number being squared. The 2 is called an exponent. For example, 5 x 5 can be written as 52, which is read as “5 squared.” The sequence of squares is       12 22 32 42 52 or, equivalently,       1 4 9 16 25 In about 600 B.C., mathematicians in Greece began representing numbers with dots arranged in geometric shapes. For example, the figures below show how they represented the sequence of square numbers. These figures reveal how the square numbers got their name. The square numbers are the numbers that can be represented by dots in square arrays. The number 25, for example, is a square number because it is the number of dots in a square array with 5 dots per side. In addition to calling 25 the square of 5, we call 5 a square root of 25. In symbols, 25 = 52 and 5 = √25. QUESTIONS The first six terms in the sequence of squares are       1 4 9 16 25 36 Copy these six terms into the 2nd column of the table and continue the sequence by writing the next six terms. Find the difference between each pair of consecutive terms in the table above. Enter these differences in the 3rd column under the Delta symbol. What do you notice about the resulting sequence? What sequence do they make? What is the common difference between each pair of consecutive terms in the new sequence? Write this difference in the 4th column. What are the partial sums of the first 12 odd numbers? What sequence do they make?       1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 (Partial sums are the cumulative totals of the terms in a series.) Inductively, what does the sum of the first n odd numbers seem to equal? What is the nth term of the first n even numbers? What is the nth term of the first n odd numbers? If 2n-1 is an odd number, write the next odd number as an expression of n. If the sum of the first n odd numbers is n2, guess the sum of the first n+1 odd numbers. But is your guess always true? If it is, then adding the next odd number (2n+1) to n2 should equal the n+1 term for any number n. Prove that if 1 + 3 + 7 + ... + 2n-1 = n2 so that 1 + 3 + 7 + ... + 2n-1 + (2n+1) = n2 + (2n+1) then 1 + 3 + 7 + ... + 2n-1 + (2n+1) = (n+1)2 When you prove that n2 + (2n+1) = (n+1)2, you will prove your guess is true for any number n. Why? If termn implies your guess is true for termn+1, then termn+1 implies that termn+2 is also true, and so on forever... AREAS & PERIMETERS The area of a square is the number of square units inside. The figures above illustrate the areas of the same squares. Copy and complete the following number sequence of the areas of these squares.       1 4 ? ? ? What kind of sequence do the areas form? These figures illustrate a sequence of squares in which the length of the side is successively doubled. What are the perimeters of these four squares? What happens to the perimeter of a square if the length of its side is doubled? What are the areas of these four squares? What happens to the area of a square if the length of its side is doubled? TRIANGULAR NUMBERS At the beginning of a game of pool, the 15 numbered balls are arranged in the triangular pattern shown here. For this reason, the number 15 is called a triangular number. The first terms in the sequence of triangular numbers are illustrated by the figures below. Write the five numbers illustrated and continue the sequence to show the next five terms. Copy and complete the following pattern. 1 = 1 1 + 2 = ? 1 + 2 + 3 = ? 1 + 2 + 3 + 4 = ? 1 + 2 + 3 + 4 + 5 = ? What kind of numbers are the numbers in the right column? Write the first 10 triangular numbers in the table below and then add each pair of consecutive numbers as shown below. What do you notice about the resulting sequence? What sequence does the right column make? What connection between the square numbers and the triangular numbers do the figures below illustrate? Hint: look in your table above. PACKING SQUARES How many squares does a chessboard contain? (Such questions are useful when packing objects for shipment.) How many are 1x1 squares? How many are 2x2 squares? How many are 3x3 squares? How many are 4x4 squares? How many are 5x5 squares? How many are 6x6 squares? How many are 7x7 squares? How many are 8x8 squares? What is the sum of all the squares? Add the answers from the previous 8 questions. Fill in the table below by computing n(n+1)(2n+1)/6 for each n from 1 to 10. What does the last number in the n(n+1)(2n+1)/6 column represent (in 10 words or less)? Hint: Fill in the Δ column with the differences between consecutive numbers in the column labelled n(n+1)(2n+1)/6 . How many cannonballs can be stacked on a base of 8x8 cannonballs? (Such questions are useful when stacking objects such as bowling balls and soup cans.) How many cannonballs can be stacked on a base of 10x10 cannonballs? You may use induction or deduction.

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From wHolt - 12/9/08 5:48 PM

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