Summing Multiples

Kim Seay

EMAT6680

This exploration begins by asking students to sum the multiples of 3 from 1 - 300, and 7 from 1 - 700. The challenge is to find a pattern for the sum of the multiples of any whole number (n) from 1 - 100*n. This is easy to do with a spreadsheet.

 

1 3 7 10 20
2 6 14 20 40
3 9 21 30 60
4 12 28 40 80
5 15 35 50 100
6 18 42 60 120
7 21 49 70 140
8 24 56 80 160
9 27 63 90 180
10 30 70 100 200
11 33 77 110 220
12 36 84 120 240
13 39 91 130 260
14 42 98 140 280
15 45 105 150 300
16 48 112 160 320
17 51 119 170 340
18 54 126 180 360
19 57 133 190 380
20 60 140 200 400
21 63 147 210 420
22 66 154 220 440
23 69 161 230 460
24 72 168 240 480
25 75 175 250 500
26 78 182 260 520
27 81 189 270 540
28 84 196 280 560
29 87 203 290 580
30 90 210 300 600
31 93 217 310 620
32 96 224 320 640
33 99 231 330 660
34 102 238 340 680
35 105 245 350 700
36 108 252 360 720
37 111 259 370 740
38 114 266 380 760
39 117 273 390 780
40 120 280 400 800
41 123 287 410 820
42 126 294 420 840
43 129 301 430 860
44 132 308 440 880
45 135 315 450 900
46 138 322 460 920
47 141 329 470 940
48 144 336 480 960
49 147 343 490 980
50 150 350 500 1000
51 153 357 510 1020
52 156 364 520 1040
53 159 371 530 1060
54 162 378 540 1080
55 165 385 550 1100
56 168 392 560 1120
57 171 399 570 1140
58 174 406 580 1160
59 177 413 590 1180
60 180 420 600 1200
61 183 427 610 1220
62 186 434 620 1240
63 189 441 630 1260
64 192 448 640 1280
65 195 455 650 1300
66 198 462 660 1320
67 201 469 670 1340
68 204 476 680 1360
69 207 483 690 1380
70 210 490 700 1400
71 213 497 710 1420
72 216 504 720 1440
73 219 511 730 1460
74 222 518 740 1480
75 225 525 750 1500
76 228 532 760 1520
77 231 539 770 1540
78 234 546 780 1560
79 237 553 790 1580
80 240 560 800 1600
81 243 567 810 1620
82 246 574 820 1640
83 249 581 830 1660
84 252 588 840 1680
85 255 595 850 1700
86 258 602 860 1720
87 261 609 870 1740
88 264 616 880 1760
89 267 623 890 1780
90 270 630 900 1800
91 273 637 910 1820
92 276 644 920 1840
93 279 651 930 1860
94 282 658 940 1880
95 285 665 950 1900
96 288 672 960 1920
97 291 679 970 1940
98 294 686 980 1960
99 297 693 990 1980
100 300 700 1000 2000
5050 15150 35350 50500 101000

Once students have experimented with different values of "n", it becomes clear that the some of the multiples will be 5000n+50n or 5050n.

This can be proven by finding the multiples of n from 1 - 100*n. (n, 2n, 3n, 4n, 5n, 6n...96n, 97n, 98n, 99n, 100n). The sum of these numbers will always be 5050n.

Conclusion:

This is a good exploration to familiarize students with using spreadsheets. What might take an extremely long time to realize using a calculator, can be done in minutes in Microsoft Excel. It also allows the opportunity to point out to students that simply using different values of n in the spreadsheet does not prove that this will work for any integer. Students benefit from the practice of constructing a spreadsheet as well as determining how they could prove their theory.

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