1. The natural odd numbers are arranged as follows:
Line 1:
1

Line 2:

3, 5

Line 3:

7, 9, 11

Line 4:

13, 15, 17, 19

a) Write the numbers on the seventh line.

b) Find the first and last number on the 101st line.

c) Show that the sum of the numbers in the first 101 lines is a perfect square.
Subject 1, author. ***

The natural odd numbers are arranged as follows:
Line 1:
1
Line 2:
3, 5
Line 3:
7, 9, 11

a) Write the numbers on the seventh line.

b) Find the first and last element on the 101st line.

c) Show that the sum of the elements in the first 101 lines is a perfect square.

| Details of solution | Associated score |
| :--- | :---: |
| a) The first 6 lines contain $1+2+\ldots+6=21$ odd numbers. The 7th line is 43, 45, ... 55 | 1 |
| b) The first 100 lines contain $(1+2+3+\ldots+100)=5050$ numbers | 1 |
| The 101st line starts with the 5051st odd number $2 \cdot 5050+1=10101$ | 2 |
| The 101st line ends with $2 \cdot 5150+1=10301$ | 1 |
| c) $1+3+5+\ldots+10301=10302 \cdot 5151: 2=5151 \cdot 5151=5151^{2}$ | 2 |