124
Proofs, Induction, and Number Theory
20. 1 # 2 # 3 + 2 # 3 # 4 + c + n(n + 1)(n + 2) =
n(n + 1)(n + 2)(n + 3)
4
21.
1
1 # 4
+
1
4 # 7
+
1
7 # 10
+ c +
1
(3n − 2)(3n + 1)
=
n
3n + 1
22. 1 # 1! + 2 # 2! + 3 # 3! + c + n # n! = (n + 1)! − 1 where n! is the product of the positive integers
from 1 to n.
23. 1 + 4 + 4
2
+ c + 4
n
=
4
n+1
− 1
3
24. 1 + x + x
2
+ c + x
n
=
x
n+1
− 1
x − 1
where x is any integer > 1
25. 1 + 4 + 7 + 10 + c + (3n − 2) =
n(3n − 1)
2
26. 1 + 3x # 2 − (x − 1)4 + 3x # 3 − (x − 1)4 + c + 3x # n − (x − 1)4 =
n 3xn − (x − 2)4
2
where x is any integer ≥ 1
27. A geometric progression (geometric sequence) is a sequence of terms where there is an initial term a and
each succeeding term is obtained by multiplying the previous term by a common ratio r. Prove the formula
for the sum of the first n (n ≥ 1) terms of a geometric sequence where r ∙ 1:
a + ar + ar
2
+ c + ar
n−1
=
a − ar
n
1 − r
28. An arithmetic progression (arithmetic sequence) is a sequence of terms where there is an initial term a and
each succeeding term is obtained by adding a common difference d to the previous term. Prove the formula
for the sum of the first n (n ≥ 1) terms of an arithmetic sequence:
a + (a + d) + (a + 2d) + c + 3a + (n − 1)d4 =
n
2
32a + (n − 1)d4
29. Using Exercises 27 and 28, find an expression for the value of the following sums.
a. 2 + 2 # 5 + 2 # 5
2
+ c + 2 # 5
9
b. 4 # 7 + 4 # 7
2
+ 4 # 7
3
+ c + 4 # 7
12
c. 1 + 7 + 13 + c + 49
d. 12 + 17 + 22 + 27 + c + 92
30. Prove that
(−2)
0
+ (−2)
1
+ (−2)
2
+ c + (−2)
n
=
1 − 2
n+1
3
for every positive odd integer n.
31. Prove that n
2
> n + 1 for n ≥ 2.
32. Prove that n
2
≥ 2n + 3 for n ≥ 3.
33. Prove that n
2
> 5n + 10 for n > 6.
34. Prove that 2
n
> n
2
for n ≥ 5.
In Exercises 35–40, n! is the product of the positive integers from 1 to n.
35. Prove that n! > n
2
for n ≥ 4.