Discrete Math

2. Write the first four terms of the sequence defined by bj = 1 + 2j,
for all integers j ( 0.

b0 =



b1 =



b2 =



b3 =

14. Find an explicit formula for the sequence with the given initial
terms: 1/4, 2/9, 3/16, 4/25, 5/36, 6/49

23. Compute the summation.

=

27. Write the summation in expanded form.

=

30. Write using summation notation: (13 – 1) + (23 – 1) + (33 –
1) + (43 – 1)

49. 5!/7! =

(Be careful to use parentheses where necessary.)

53. n!/(n ( 2)! = __________________________________________________

(Be careful to use parentheses where necessary.)

NOTES: I have italicized the variable n (since the Equation Editor
normally does this automatically), but it is not necessary. Plain n is
fine.

6. (Required) Without using Theorem 4.2.2, use mathematical induction
to prove that

P(n): 2 + 4 + 6 + ...+ 2n = n2 + n for all integers n ( 1.

7. (Required) Without using Theorem 4.2.2, use mathematical induction
to prove that

P(n): 1 + 5 + 9 + ... + (4n ( 3) = n(2n ( 1) for all integers n ( 1

Prove the statements in #9-12 by mathematical induction.

for all integers n (1.

10. (Required) P(n): 13 + 23 + ... + n3 = [n(n + 1)/2]2

for all integers n ( 1.

for each integer n with n ( 2.

20. (Required) Use the formula for the sum of the first n integers
and/or the formula for the sum of a geometric sequence to find the sum 5
+ 10 + 15 + 20 + ... + 300.

21. Find an explicit formula for the sequence with the initial terms

2, 6, 12, 20, 30, 42, 56

22. Write using summation notation: