Patterns, Parity, and Consecutive Numbers

Sums of Consecutive Numbers

Can every number be written as a sum of consecutive natural numbers? Let’s verify.

• $3 = 1 + 2$
• $5 = 2 + 3$
• $6 = 1 + 2 + 3$
• $10 = 1 + 2 + 3 + 4$

The 4-Number Sign Puzzle

Take any 4 consecutive numbers, say $3, 4, 5, 6$. Place + or - signs between them. Example: $3 + 4 - 5 + 6 = 8$ $3 - 4 - 5 - 6 = -12$

Observation: The result is always EVEN.

Algebraic Proof

Let the four numbers be $a, b, c, d$. Consider the expression $S = a + b + c + d$. If we change the sign of $b$ from $+$ to $-$, the new sum is $S' = a - b + c + d$. The difference is $S - S' = 2b$. Since $2b$ is always an even number, changing a sign changes the total sum by an even amount. Therefore, if the initial sum is even (or odd), all variations will maintain the same parity.

For 4 consecutive integers (e.g., $n, n+1, n+2, n+3$): Sum $= 4n + 6 = 2(2n + 3)$. Since $2(2n+3)$ is a multiple of 2, the sum is always even. Thus, any variation of signs will also result in an even number.

Pairs to Make Fours (Multiples of 4)

When do two even numbers sum to a multiple of 4? Even numbers are of two types:

1. Multiples of 4 ($4k$): $4, 8, 12 \dots$ (Remainder 0)
2. Not Multiples of 4 ($4k+2$): $2, 6, 10 \dots$ (Remainder 2)

Visualizing the Logic

Conclusion:

• Sum of two multiples of 4 $\rightarrow$ Multiple of 4.
• Sum of two non-multiples of 4 (even) $\rightarrow$ Multiple of 4 (because $2+2=4$).