Memorial University of Newfoundland Memorial University of Newfoundland

Department of Mathematics and Statistics

Mathematical Challenge for Junior High School Students 1999-2000: # 2

Grades 7, 8 and 9

An arithmetic sequence is a sequence of terms where the difference between successive terms is a constant amount. For example:

2,  3,  4,  ¼  12,  13;
      
5,  9,  13  ¼  49,  53;
      
101,  202,  ¼  707,  808.

The differences are: 1 in the first example, 4 in the second example and 101 in the third example. Here, we only permit positive differences and we require the first term to be positive.

The number, 20 001 999 can be written as the sum of the terms of an arithmetic sequence. This can be done in several different ways. For example:

20 001 999
=
10 001 000+10 000 999   =   6 667 332+6 667 333+6 667 334

Find as many different examples of sums of arithmetic sequences adding up to 20 001 999 as you can. Note that ``different'' means that you cannot give the same numbers in a different order.

Give your solution to your Mathematics teacher, who will forward it to:

Mathematical Challenge for Junior High School Students 1999-2000: # 2
Department of Mathematics and Statistics
Memorial University of Newfoundland
St. John's, Newfoundland. A1C 5S7

DEADLINE for postmarking of solutions: 2 December 1999
Make sure that your solution is written legibly and neatly.

A copy of the following form MUST accompany the entry for each student:

STUDENT'S NAME ------------------------------------- (Please PRINT)
School -------------------------------------
Community -----------------------------
Grade --------- Teacher's Name -----------------------------
This is my own unaided work (Calculators are permitted,
but computers are not)
Signed ------------------------------------- Date -------------------