Do these patterns occur no matter which 4 consecutive numbers are chosen? Is there a way Class 8

Do these patterns occur no matter which 4 consecutive numbers are chosen? Is there a way Class 8

Question 1.

Do these patterns occur no matter which 4 consecutive numbers are chosen? Is there a way to find out through reasoning?

Hint : Use algebra and describe the 8 expressions in a general form. Class 8

Solution:

Yes.

Suppose a, b, c, d are four consecutive numbers.

Then, a + b + c + d is an even number, because in a, b, c and d, two numbers will be odd and two numbers will be even.

Now, let us examine a - b + c + d.

We have (a + b + c + d) - (a - b + c + d) = 2b, which is even.

Therefore, (a - b + c + d) will also be even.

Explanation 1: Let us consider any of the 8 expressions formed by four numbers a, b, c, and d. When one of its signs is switched, its value always increases or decreases by an even number! Let us see why.

Consider one of the expressions: a + b – c – d.

Replacing +b by – b, we get

a – b – c – d.

By how much has the number changed? It has changed by

(a + b – c – d) – (a – b – c – d)

= a + b – c – d – a + b + c + d (notice how the signs changed when we opened the second set of brackets)

= 2b (this is an even number).

If the difference between two numbers is even, can they have different parities? No! So either both are even or both are odd.

Now, let us see what happens when a negative sign is switched to a positive sign.