5. The sum of the first five terms of an arithmetic sequence is 50, and the difference between the fifth and the second term is 9. Calculate which term is seven times as large as the first.

## You have 120 minutes to solve.

Solve each problem independently on the provided paper, with each problem on a separate page. Do not sign the sheets; write only your code. Write your answers in ink, clearly and neatly. Draw graphs of functions with a pencil. If you solve a problem in multiple ways, clearly indicate which solution the evaluator should grade. If you make a mistake, cross it out.

Each problem will be graded on a scale of 0 to 6 points.

The National Competition Committee wishes you great success.

## 8th Mathematics Knowledge Competition for

students

dMFA

of secondary technical and vocational schools

National Competition, April 19, 2008

## Solutions and Scoring Guide

## A competitor who arrives at the solution by any correct method (even if not anticipated by the scoring guide) receives all possible points.

A correct method is one that

- sensibly takes into account the problem statement,
- leads to the solution of the problem,
- is mathematically correct and complete.

If an intermediate or final result can be recognized, guessed, read from a diagram, or calculated mentally, the competitor generally receives all the points provided. However, if the solution is guessed (and cannot be arrived at without calculation), even if it is correct, it is scored with 0 points.

A competitor who has only partially solved the problem, with correct but incomplete steps, cannot receive more than half of the possible points.

## First Year
5. We use the formula for the sum of the first five terms of an arithmetic sequence $s_{5}=\frac{5}{2}\left(2 a_{1}+4 d\right)$. We consider that $a_{5}-a_{2}=9$ or $a_{1}+4 d-a_{1}-d=9$. We calculate the difference $d=3$. We substitute the difference into the sum formula $\frac{5}{2}\left(2 a_{1}+4 d\right)=50$ and calculate $a_{1}=4$. We write the relation $a_{n}=7 a_{1}=28$. Using the known data, we calculate that the ninth term is seven times the first.

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-11.jpg?height=60&width=1636&top_left_y=2043&top_left_x=210)

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-11.jpg?height=49&width=1636&top_left_y=2100&top_left_x=210)

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-11.jpg?height=49&width=1639&top_left_y=2143&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-11.jpg?height=57&width=1639&top_left_y=2193&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-11.jpg?height=54&width=1636&top_left_y=2246&top_left_x=210)

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-11.jpg?height=51&width=1639&top_left_y=2296&top_left_x=206)