When is P a Perfect Square?

P = 1 + (2 · 3) + (4 · 5 · 6) + (7 · 8 · 9 · 10) + (11 · 12 · 13 · 14 · 15) + …

Starting with 1, the first term, we can add as many terms as we wish.

How many terms should we add, such that P is a perfect number?

Answer

Starting with the 3rd term, each term includes an even number and a multiple of 5, therefore it is a multiple of 10.

So, when we add all the terms starting with the 3rd term, what digit is in the units place of the sum? Zero, because we are adding a bunch of multiples of 10.

Now let’s add all the terms of P, including the first two terms, 1 and 2·3.

What is the last digit of P? It has to be the last digit of the first two terms added: 1 + 2·3 = 7, since the rest of the digits add up to a number whose last digit is 0 (zero).

The last digit of a perfect square can only be 0, 1, 4, 5, 6, or 9. A number ending in 7 can not be a perfect square. Does this mean that P can never be a perfect square?

P can be a perfect square but only if we are including its first term, namely 1.

P = 1 is a perfect square.

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