Let's analyze the number pattern to answer parts c and d.

Given Number Pattern:

1
2 3 4
5 6 7 8 9

Identifying the Pattern:

• The first line has 1 number.
• The second line has 3 numbers.
• The third line has 5 numbers.

The pattern increases by 2 numbers in each subsequent line.

General Formula:

To find out how many numbers are in the nth line, we notice the pattern:

• 1st line: $1$ number (1 number)
• 2nd line: $1 + 2 = 3$ numbers
• 3rd line: $3 + 2 = 5$ numbers
• and so on...

So, for the nth line, the number of elements is given by:
$\text{Number of elements in the nth line} = 2n - 1$

c. What will be the last number in the 9th line?

To find the last number on the 9th line, we need to add up all the numbers in the previous lines and add the elements in the 9th line:

The sum of the first $n$ odd numbers (which is what we're dealing with here) is $n^2$. So, to find the last number on the 9th line:

• The last number on the 9th line will be the sum of the first 9 lines, which is $9^2 = 81$.

So, the last number on the 9th line is 81.

d. Write the first and last numbers of the 10th line.

Since the 9th line ends with 81, the first number on the 10th line will be the next number, which is 82.

To find the last number on the 10th line, we know the 10th line has $2 \times 10 - 1 = 19$ numbers. Adding 19 to the first number on the 10th line (82):

• The last number on the 10th line = $82 + 18 = 100$.

So, the first number on the 10th line is 82, and the last number is 100.

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Summary:

c. The last number in the 9th line is 81.
d. The first and last numbers of the 10th line are 82 and 100.

Would you like any more details or have any questions?

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Here are 5 related questions you might find interesting:

1. What is the sum of all the numbers in the 9th line?
2. How many numbers are there in the 12th line?
3. What is the difference between the last numbers of the 8th and 9th lines?
4. How would the sequence continue if you wanted the 15th line?
5. What is the general formula for the first number in any line?

Tip: When dealing with number patterns, identifying the rule governing the sequence helps in predicting future terms efficiently.