Name ________________________
Worksheet 1.6
More proof by mathematical induction-algebraic formulas.
Definitions.
Recursively defined sequence. A function with domain the natural numbers (which is usually called a
sequence) is defined recursively when one is told how to use each value to find the next. For example
consider the sequence 3, 5, 7, 9, ...
Then F(1) = 3, F(n+1)=F(n)+2 defines the sequence recursively.
Explicitly defined sequence. A sequence is defined explicitly when one is told how to find each value
directly without needing to know previous values.
An explicit definition of the same sequence is F(n)=2n+1.
Sigma Notation. In mathematics the Greek letter for S stands for "sum' and a couple of examples
suffice to explain the notation.
5
 k  0  1 2  3  4  5  15
Say "the sum from k = 0 to 5 of k."
k0
5
9
i3
j5
(2i  3)  (2  3  3) (2  4  3)  (2  5  3)  33 2  2  2  2  2  2  10

This notation is only used with whole numbers as variables.
The notation is just shorthand for addition, so sums within a sum can be separated and, by the


distributive law, factors can be factored out. Here is an example:
5
(2q
2
5
5
5
q1
q1
 3q  1)  2q  3q  1
q1
2
q1
Problems
We will do only afew of these. The rest are for your interest or can be done for extra credit.
1. Practice with sigma notation. Evaluate these sums:
4
a.
 3i
7
2
b.
i0
(1)
j0
n
2. 1 2  3  4  ... n   i 
i1
n(n  1)
2
j
24
c.
4
r0
(Sum of the first n integers.)
n


1  3  5  7...(2n 1)   2i 1 n 2 
i1
3.
(Sum of the first n odd integers.)
n

4.
2  4  6  8...2n   2i n(n 1)
i 1
(Sum of the first n even integers.)
5.
n(n  1)(2n  1)
6
12  2 2  32 ... n2 
(Sum of the first n squares.)
More Induction Practice
Do Problems 5 and 8 first then time permitting try 1. and 15.
The others are left for your delight and edification!
1. Find a rule for the sum of cubes 13+23+33+...+n3.
2. 1  2  4  8  16...2
n1
 2 n 1
(Sum of powers of 2.)
3. The sum of three successive cubes is always divisible by 9.
1 2 1
n  n 1
2
4. 2
is a whole number for all n.
n
n
5. 17 14 is divisible by 3.
n
n
6. 11  5 is divisible by 16 if n is odd and not divisible by 16 if n is even.
n
7. n!  2 for n  4 .
2n
8. 3 is a factor of 2  1
9. x – y is a factor of xn – yn..
2n-1
2n-1
10. x + y is a factor of x + y .
1 1 1
1
1
1
1
  



3
11. Here's a striking fact. 1 2 3 12 13 2  3 1 2 3
. Here one is adding up the
reciprocals of the products of the members of every non-empty subset of the first n numbers. When this
is done for 3 the sum is 3. Prove the result is always n.
12. You probably know a test for divisibility by 3. Add up the digits and see if that number is divisible
by three. A test for divisibility by 11 is to alternately add and subtract the digits of the number from left
to right. It the result is divisible by 11 so is the original number
13. 5 is a factor of 2
2n1
 32n1 .
14. n3+3n2+2n is divisible by 3.
15. Suppose n people all have a unique piece of gossip to share. They start phoning each other. Assume
that when each person speaks with someone he or she shares all the gossip he or she knows up to that
time. What is the minimum number of phone calls the group must make so that everyone knows all the
gossip. Prove your assertion.