Sum of Consecutive Numbers

How many positive integers from 1 to 100 inclusive can be written as the sum of two or more consecutive positive integers?  (Different examples 10 and 51: 10 = 1 + 2 + 3 + 4 and 51 = 25 + 26)
Source: mathcontest.olemiss.edu 3/7/2011

SOLUTION
Sum of two consecutive numbers
1+2=3
2+3=5
3+4=7
4+5=9
\cdots
48+49=97
49+50=99
  
There are 100 numbers from 1 to 100. Half of them are even and half are odd. All the above sums are odd. Thus, there are 49 numbers from 3 to 99 that can be written as the sum of two consecutive positive integers.
  
Sum of three consecutive integers
1+2+3=6
2+3+4=\underline{9}        duplicates are indicated by underlines.
3+4+5=12
4+5+6=\underline{15}
5+6+7=18
6+7+8=\underline{21}
\cdots
29+30+31=90
30+31+32=\underline{93}
31+32+33=96
32+33+34=\underline{99}
 
  
An easy way to count these sums is to think of them as multiples of 6:
6=6\times 1
12=6\times 2
18=6\times 3
\cdots
90=6\times 15
96=6\times 16
  
Thus, sixteen numbers {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96} can be written as the sum of three consecutive positive integers.
 
Sum of four consecutive integers
1+2+3+4=10
2+3+4+5=14
3+4+5+6=\underline{18}
4+5+6+7=22
5+6+7+8=26
6+7+8+9=\underline{30}
7+8+9+10=34
8+9+10+11=38
9+10+11+12=\underline{42}
10+11+12+13=46
11+12+13+14=50
12+13+14+15=\underline{54}
13+14+15+16=58
14+15+16+17=62
15+16+17+18=\underline{66}
16+17+18+19=70
17+18+19+20=74
18+19+20+21=\underline{78}
19+20+21+22=82
20+21+22+23=86
21+22+23+24=\underline{90}
22+23+24+25=94
23+24+25+26=98
 
Sixteen numbers {10, 14, 22, 26, 34, 38, 46, 50, 58, 62, 70, 74, 82, 86, 94, 98} can be written as the sum four consecutive positive integers.

Sum of five consecutive integers
1+2+3+4+5=\underline{15}
2+3+4+5+6=20
3+4+5+6+7=\underline{25}
4+5+6+7+8=\underline{30}        Surprise! 30 is a duplicate but 20 is not.
5+6+7+8+9=\underline{35}
6+7+8+9+10=40
7+8+9+10+11=\underline{45}
8+9+10+11+12=\underline{50}
9+10+11+12+13=\underline{55}
10+11+12+13+14=\underline{60}
11+12+13+14+15=\underline{65}
12+13+14+15+16=\underline{70}
13+14+15+16+17=\underline{75}
14+15+16+17+18=80
15+16+17+18+19=\underline{85}
16+17+18+19+20=\underline{90}
17+18+19+20+21=\underline{95}
18+19+20+21+22=100
  
Four numbers {20, 40, 80, 100} can be written as the sum of five consecutive positive integers.
  
Sum of six consecutive integers
1+2+3+4+5+6=\underline{21}
2+3+4+5+6+7=\underline{27}
3+4+5+6+7+8=\underline{33}
4+5+6+7+8+9=\underline{39}
\cdots
 
Luckily, all the sums are odd and therefore duplicates.
 
Sum of seven consecutive numbers
1+2+3+4+5+6+7=28
2+3+4+5+6+7+8=\underline{35}
3+4+5+6+7+8+9=\underline{42}
4+5+6+7+8+9+10=\underline{49}
5+6+7+8+9+10+11=56
6+7+8+9+10+11+12=\underline{63}
7+8+9+10+11+12+13=\underline{70}
8+9+10+11+12+13+14=\underline{77}
9+10+11+12+13+14+15=\underline{84}
10+11+12+13+14+15+16=\underline{91}
11+12+13+14+15+16+17=\underline{98}
 
Two numbers {28, 56} can be written as the sum of seven consecutive positive integers.
 
Sum of eight consecutive integers 
1+2+3+4+5+6+7+8=\underline{36}
2+3+4+5+6+7+8+9=44
3+4+5+6+7+8+9+10=52
4+5+6+7+8+9+10+11=\underline{60}
5+6+7+8+9+10+11+12=68
6+7+8+9+10+11+12+13=76
7+8+9+10+11+12+13+14=\underline{84}
8+9+10+11+12+13+14+15=92
9+10+11+12+13+14+15+16=\underline{100}
 
Five numbers {44, 52, 68, 76, 92} can be written as the sum of eight consecutive positive integers.

Sum of nine consecutive integers
There are seven numbers {45, 54, 63, 72, 81, 90, 99} in this category but they are duplicates.

Sum of ten consecutive integers
There are six numbers {55, 65, 75, 85, 95} in this category but they are duplicates.

Sum of eleven consecutive integers
There are four numbers {66, 77, 88, 99} in this category; all are duplicates except 88. Thus, there is one number {88} that can be written as the sum of eleven consecutive positive integers.

Sum of twelve consecutive integers
There are two numbers {78, 90} in this category; both are duplicates.

Sum of thirteen consecutive integers
There is one number {91} in this category; it is a duplicate.

Sum of fourteen consecutive integers
1+2+3+\cdots+13+14=105.

The sum is greater than 100. So, we stop here.

Summary
Sum of two consecutive integers: 49 numbers
Sum of three consecutive integers: 16
Sum of four consecutive integers: 16
Sum of five consecutive integers: 4
Sum of six consecutive integers: 0
Sum of seven consecutive integers: 2
Sum of eight consecutive integers: 5
Sum of nine consecutive integers: 0
Sum of ten consecutive integers: 0
Sum of eleven consecutive integers: 1
Sum of twelve consecutive integers: 0
Sum of thirteen consecutive integers: 0

Total = 49+16+16+4+2+5+1=93

Answer: 93.

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About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , . Bookmark the permalink.

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