Problem 4. The sum of eight consecutive natural numbers is equal to 92. What are these numbers?
Solution. First method. Let the smallest of the required numbers be denoted by $x$. Then the required numbers are $x, x+1, x+2, x+3, x+4, x+5, x+6$ and $x+7$. From the condition of the problem, we have

$$
x+(x+1)+(x+2)+(x+3)+(x+4)+(x+5)+(x+6)+(x+7)=92
$$

i.e., $8 x+28=92$, from which we get $x=(92-28): 8=64: 8=8$. Therefore, the required numbers are $8,9,10,11,12,13,14$ and 15.

Second method. The sum of the first eight natural numbers is

$$
1+2+3+4+5+6+7+8=36
$$

If the first addend is replaced by the next natural number, then the sum of the new eight consecutive numbers increases by 8, and this is true for any such newly obtained eight consecutive natural numbers. Since $(92-36): 8=56: 8=7$, we conclude that the procedure must be repeated seven times, which means that the first addend must be replaced by the next natural number seven times. Thus, we obtain the following sequences of numbers

$$
\begin{aligned}
& 2,3,4,5,6,7,8,9 \\
& 3,4,5,6,7,8,9,10 \\
& 4,5,6,7,8,9,10,11 \\
& 5,6,7,8,9,10,11,12 \\
& \text { 6, 7,8,9,10,11,12,13; } \\
& 7,8,9,10,11,12,13,14 \\
& 8,9,10,11,12,13,14,15
\end{aligned}
$$

which means that the required numbers are $8,9,10,11,12,13,14$ and 15.

Third method. The second number is 1 greater than the first, and the seventh is 1 less than the eighth. Therefore, the sum of the first and eighth of the required numbers is equal to the sum of the second and seventh numbers. Similarly,

![](https://cdn.mathpix.com/cropped/2024_06_05_3a3538b5fdc92afac337g-05.jpg?height=315&width=711&top_left_y=1616&top_left_x=769)
we conclude that this sum is equal to the sum of the third and sixth numbers, as well as the sum of the fourth and fifth numbers. Therefore, we have 4 pairs of numbers with equal sums, so the sum of the numbers in each pair is equal to $92: 4=23$. However, the fourth and fifth numbers are two consecutive
numbers, so since $11+12=23$, we conclude that these are the numbers 11 and 12. Now, it is easy to see that the required numbers are 8, 9, 10, 11, 12, 13, 14 and 15.

## VI section