Task B-1.7. In the city of the future, super trains are used for the transportation of residents. They consist of cars with a different number of compartments, and each of them has a certain number of double seats. The first car has only one compartment with one double seat. To it, a car is attached which has one more compartment in which there are two more double seats than in the first compartment, and so on. Each subsequent car is increased by one compartment with two more double seats than in the previous compartment. How many passengers can fit in the $n$-th car? How many passengers can fit in the 10th car?
## Solution.

The train looks like this:

- the first carriage has one compartment with $1 \cdot 2$ seats;
- the second carriage has:

the first compartment with $1 \cdot 2$ seats, the second compartment with $1 \cdot 2+2 \cdot 2=3 \cdot 2$ seats;

(1 point)

- the third carriage has:

the first compartment with $1 \cdot 2$ seats, the second compartment with $3 \cdot 2$ seats, the third compartment with $5 \cdot 2$ seats, etc.

(1 point)

- the $n$-th carriage has:

$n$ compartments with $S_{n}=1 \cdot 2+3 \cdot 2+5 \cdot 2+7 \cdot 2+\ldots+(2 n-1) \cdot 2$ seats

$$
S_{n}=2 \cdot(1+3+5+7+\ldots+(2 n-1))
$$

The sum of the first $n$ odd natural numbers can be calculated by grouping the terms

$$
\begin{aligned}
& 1+3+5+7+\ldots+(2 n-5)+(2 n-3)+(2 n-1)= \\
= & (1+2 n-1)+(3+2 n-3)+(5+2 n-5)+\ldots+(2 n-3+3)+(2 n-1+1)= \\
= & 2 n+2 n+2 n+\ldots+2 n+2 n=2 n \cdot \frac{n}{2}=n^{2}
\end{aligned}
$$

The number of passengers in the $n$-th carriage will be $S_{n}=2 n^{2}$.

The tenth carriage has 10 compartments, and the number of passengers in that carriage is

$$
S_{10}=1 \cdot 2+3 \cdot 2+5 \cdot 2+7 \cdot 2+\ldots+(2 \cdot 10-1) \cdot 2=2 \cdot 10^2=200
$$

## Note:

Accept the calculation of the sum of the first $n$ odd numbers using the formula $\frac{n}{2}\left(a_{1}+a_{n}\right)$, but not the conclusion by incomplete induction that this sum is equal to $n^{2}$.

## SCHOOL/CITY COMPETITION IN MATHEMATICS

## 2nd grade - high school - B variant

January 27, 2014.

IF A STUDENT HAS A DIFFERENT METHOD OF SOLVING THE PROBLEM, THE COMMITTEE IS OBLIGED TO GRADE AND EVALUATE THAT METHOD APPROPRIATELY.