## Task B-1.2.

In a sequence of six natural numbers, the third and each subsequent number is equal to the sum of the two preceding ones. Determine all such sequences of numbers, if the fifth number in the sequence is equal to 25.
## Solution.

Let $a$ and $b$ be the first and second numbers in the given sequence. Then the third number in the sequence is $a+b$, and the fourth is $a+2b$.

The number 25 is equal to the sum of the third and fourth numbers in the sequence, that is,

$(a+b) + (a+2b) = 25$. Thus we get the equation

$2a + 3b = 25$
$2a = 25 - 3b \left(^{*}\right)$

Since the left side is an even and positive number, the right side must also be.

Therefore, $b$ is an odd number less than 8, that is, one of the numbers 1, 3, 5, or 7.

For these values of $b$, we get from (*) that the number $a$ is equal to 11, 7, 5, or 2, respectively.

The required sequences of numbers are:

$11, 1, 12, 13, 25, 38$

$8, 3, 11, 14, 25, 39$

$5, 5, 10, 15, 25, 40$

$2, 7, 9, 16, 25, 41$.

1 point