Magnificent Mistakes in Mathematics by Alfred S. Posamentier

Author:Alfred S. Posamentier
Language: eng
Format: epub
Publisher: Prometheus Books
Published: 2013-07-10T16:00:00+00:00

From the seven examples, one could easily form the following conclusion, namely that for the natural number n, the following should hold:

1n + 13n + 28n + 70n + 82n + 124n + 139n + 151n= 4n + 7n + 34n + 61n + 91n + 118n + 145n + 148n.

These values are shown in the table below:

n Sums

0 8

1 608

2 70,076

3 8,953,712

4 1,199,473,412

5 165,113,501,168

6 23,123,818,467,476

7 3,276,429,220,606,352

To make this generalization would be a predictable behavior. However, at the same time, it would also be a marvelous mistake. This mistake does not manifest itself until we take the next case, namely, where n = 8.

Notice the two sums that we get are no longer the same:

18 + 138 + 288 + 708 + 828 + 1248 + 1398 + 1518

= 468,150,771,944,932,292.

However, 48 + 78 + 348 + 618 + 918 + 1188 + 1458 + 1488 = 468,087,218,970,647,492. As a matter of fact, the difference between these two sums is:

468,150,771,944,932,292 – 468,087,218,970,647,492 = 63,552,974,284,800.

As n increases, so does the difference between the two sums. For n = 20, the difference is 3,388,331,687,715,737,094,794,416,650,060,343,026,048,000. Therefore, to avoid such mistakes, one must be sure to prove a generalization before accepting it inductively.

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