Section 1. Getting Started: Maple as a Calculator

6. Use the sequence operator, $, to make the following sequences.

a. 2, 4, 6, 8, 10, 12

b. 20, 40, 60, ... , 260      Hint.  20 + 20*k $ k=

c. The 30 prime numbers starting with 11 and ending with 139. Hint. Use ithprime

7.* Add the numbers in the sequence in 6 c.. Hint. If you just did 6 c, then execute add(k,k=%).

> ithprime(k) $ k=5..33;

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137

> add(k,k=%);

1971

10. Use seq to obtain the sequence of the cubes of the numbers in the list [2, 5, 6, 9, 12, 44]. Add the numbers in the sequence of cubes and then obtain the prime factorization of the sum.

What do you notice about the prime factorization? (Hint. The integer 87990 is called "square free".)

11.* Obtain the prime factorization of the product of the integers in the sequence of cubes described in Exercise 10.

> seq( k^3, k=[2,5,6,9,12,44]);

8, 125, 216, 729, 1728, 85184

> mul(k,k=%);
ifactor(%);

23178378313728000

``(2)^18*``(3)^12*``(5)^3*``(11)^3

12. Use the factor procedure to factor the following polynomial expressions.

a. x^3 - x^2 + x - 1

b. x^7 - x^6 + x^5 - ... -1    Hint. Enter this as -add( (-x)^k,k=0..7) then execute factor(%).

15.* Use solve to obtain the zeros of the polynomials in Exercise 12.

> x^3 - x^2 + x - 1;
solve(%);

x^3-x^2+x-1

1, I, -I

> -add((-x)^k,k=0..7);
solve(%);

-1+x-x^2+x^3-x^4+x^5-x^6+x^7

1, I, -I, 1/2*2^(1/2)+1/2*I*2^(1/2), -1/2*2^(1/2)+1/2*I*2^(1/2), -1/2*2^(1/2)-1/2*I*2^(1/2), 1/2*2^(1/2)-1/2*I*2^(1/2)

16.* Add the zeros of the polynomial in 12 a and the zeros of the polynomial in 12 b.

> solve(x^3 - x^2 + x - 1);
add(k,k=%);

1, I, -I

1

> solve(-add((-x)^k,k=0..7));
add(k,k=%);

1, I, -I, 1/2*2^(1/2)+1/2*I*2^(1/2), -1/2*2^(1/2)+1/2*I*2^(1/2), -1/2*2^(1/2)-1/2*I*2^(1/2), 1/2*2^(1/2)-1/2*I*2^(1/2)

1