Sums of Consecutive Integers
and
CCSS Mathematical Practices
Gary Talsma
Mathematics and Statistics Department
Calvin College
Grand Rapids, MI 49546
()
Math in Action GVSU
February 9, 2013
The Problem(s) That Haunted Me
• From an activity a colleague did with elementary school teachers:
For each of the numbers from 1-100, see how many ways you can express the number as a sum of two or more consecutive positive integers.
• Problem 5 on Part II of the 2011 MMPC:
Say that a number N is a nontrivial sum of consecutive positive integers if N can be expressed as the sum of 2 or more consecutive positive integers. Determine, with proof, the set of all integers N between 1000 and 2000 which are not nontrivial sums of consecutive integers.
• A “mathematical project” from Exploring the Real Numbers by Frederick W. Stevenson:
The consecutive index of a natural number N, denoted by c(N), is the number of ways that N can be written as the sum of 2 or more consecutive positive integers.
a) Find c(N) for 2 ≤ N ≤ 50.
b) Describe the nature of the numbers whose consecutive index is k for k = 0, 1, 2, 3.
c) Find a formula for c(N) for all natural numbers N.
d) Which natural number less than or equal to 100 has the greatest consecutive index?
e) Find the smallest N so that c(N) = 100.
The Common Core State Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Looking for Patterns, Structure, and Regularity
• Describe the numbers that can be expressed as the sum of 2 consecutive positive integers.
• Describe the numbers that can be expressed as the sum of 3 consecutive positive integers.
• Describe the numbers that can be expressed as the sum of 4 consecutive positive integers.
• Describe the numbers that can be expressed as the sum of 5 consecutive positive integers.
• Describe the numbers that can not be expressed as the sum of 2 or more consecutive positive integers.
Triangular Numbers, and a Theorem
The nth triangular number Tn is the sum of the integers from 1 through n. (Sometimes we define T0 = 0.)
Tn = n (n + 1) / 2.
Theorem: For N a natural number, if N can be expressed as a sum of 2 or more consecutive positive integers, then N has an odd factor.
Proof: Suppose N can be expressed as a sum of 2 or more consecutive positive integers. Then for some integers j and k (with k > j), N = Tk – Tj = … (use the definition of triangular numbers, do some algebra, including factoring, and consider various cases for whether j and k are even or odd)
What’s the Deal with Odd Factors?
Let’s use N = 45 as an example. The odd factors of 45 are: 1, 3, 5, 9, 15, and 45.
Consider the odd factor d = 3 = 2(1) + 1. Its “partner” is 15, since 3 x 15 = 45.
45 = 3 x 15 = 15 + 15 + 15 = 14 + 15 + 16, a consecutive sum of length 3.
Let’s try that again with d = 5 = 2(2) + 1.
45 = 5 x 9 = 9 + 9 + 9 + 9 + 9 = 7 + 8 + 9 + 10 + 11, a consecutive sum of length 5.
How about with d = 9 = 2(4) + 1?
45 = 9 x 5 = 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9, a consecutive sum of length 9.
Might as well try d = 1 = 2(0) + 1.
45 = 1 x 45 = 45, a “consecutive sum” of length 1, but we’re after sums of 2 or more consecutive positive integers. Doesn’t look like 1 is a helpful odd factor.
Try d = 15 = 2(7) + 1.
45 = 15 x 3 = 3+3+3+3+3+3+3+3+3+3+3+3+3+3+3
= –4 + –3 + –2 + –1 + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 (why is this “bad”?)
= 5 + 6 + 7 + 8 + 9 + 10, a consecutive sum of length 6.
Finally, for d = 45 = 2(22) + 1, can you see what will happen?
45 = 45 x 1 = 1 + 1 + 1 + … + 1 + 1 (forty-five ones added together).
When we “redistribute” here, how low will we go for our first term? How high for the last term?
How much will “cancel”, and what will be left?
This example “suggests” that every odd factor of N, except 1, will generate a representation of N as a sum of 2 or more consecutive positive integers!
Counting the Odd Factors of a Number
What are the factors of 12?
What’s the prime factorization of 12?
How is each factor related to the prime factorization?
Since 12 = 22 x 31, every factor of 12 will be some combination of 2s and 3s.
Number of factors of 12 = (Number of choices for 2s) x (Number of choices for 3s)
How many odd factors does 12 have?
How many odd factors does 45 have?
Got It??
What’s the prime factorization of 2013?
How many odd factors does 2013 have?
In how many ways can 2013 be represented as a sum of 2 or more consecutive positive integers?
Find all the consecutive sums for 2013.