MATH117 Homework 6: due Friday, 14 October
Remember to justify your answers!
(1) Find an explicit formula for the following sequence:
1 2 3 4 5 6
0, − , , − , , − , , . . .
2 3 4 5 6 7
(2) Compute the following:
3
X
1
(a)
2m
(b)
m=0
5
Y
k2
k=2
(3) Write each of the following in either summation or product notation.
(a) (22 − 1)(32 − 1)(42 − 1)
(b) n +
n−1
2!
+
n−2
3!
+
n−3
4!
1
+ . . . n!
(4) Prove that if p is a prime number and r is an integer with 0 < r < p, then
by p.
p
r
is divisible
(5) Prove the following statement (for all positive integers n) using mathematical induction.
n
X
i=1
i3 =
n(n + 1)
2
2
(6) Prove the following statement (for all integers n ≥ 2) using mathematical induction.
n
Y
1
n+1
(1 − 2 ) =
i
2n
i=2
(7) Use mathematical induction, the product rule from calculus, and the facts that d(x)
dx = 1
d(xn )
k
k−1
n−1
and x = x · x
to prove that for all integers n ≥ 1, dx = nx
.
(8) Compute the following sum, using the formula for the sum of the first n positive integers.
(We found that formula when we found the formula for the handshake numbers.)
5 + 10 + 15 + 20 + . . . + 300
(9) Use the formula for the sum of a geometric sequence to evaluate the following sum.
32 + 33 + 34 + . . . 3n
(10) Find a formula in terms of a, r, m, and n for the sum
arm + arm+1 + arm+2 + . . . arm+n ,
where m and n are integers, n ≥ 0, and a and r are real numbers.