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\begin{tabular}{|lll|}
\hline
2 & [tex]$=2=1 \cdot 2$[/tex] \\
\hline
[tex]$2+4$[/tex] & [tex]$=6=2 \cdot 3$[/tex] \\
\hline
[tex]$2+4+6$[/tex] & [tex]$=12=3 \cdot 4$[/tex] \\
\hline
[tex]$2+4+6+8$[/tex] & [tex]$=20=4 \cdot 5$[/tex] \\
\hline
[tex]$2+4+6+8+10$[/tex] & [tex]$=30=5 \cdot 6$[/tex] \\
\hline
\end{tabular}

Look at the table. Make a conjecture about the sum of the first 10 positive even numbers.

A. The sum is [tex]$11 \times 12$[/tex].
B. The sum is [tex]$9 \times 10$[/tex].
C. The sum is [tex]$10 \times 10$[/tex].
D. The sum is [tex]$10 \times 11$[/tex].



Answer :

GinnyAnswer
To find the sum of the first 10 positive even numbers and verify which conjecture matches, let's work through the process step by step.

1. Identify the first 10 positive even numbers:
- The first 10 positive even numbers are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.

2. Calculate the sum of these numbers:
- Let's add these numbers together:
[tex]\[ 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 \][/tex]
- Grouping step-by-step to make it simpler:
[tex]\[ (2 + 20) + (4 + 18) + (6 + 16) + (8 + 14) + (10 + 12) \\ = 22 + 22 + 22 + 22 + 22 \\ = 5 \times 22 = 110 \][/tex]
So, the sum of the first 10 positive even numbers is 110.

3. Examine the conjecture choices provided:
- From the conjecture choices:
- 11 [tex]\(\times\)[/tex] 12 = 132
- 9 [tex]\(\times\)[/tex] 10 = 90
- 10 [tex]\(\times\)[/tex] 10 = 100
- 10 [tex]\(\times\)[/tex] 11 = 110

4. Determine which conjecture matches the calculated sum:
- We have found that the sum is 110. Comparing this with the conjecture choices, we see that the correct match is:
[tex]\[ 10 \times 11 = 110 \][/tex]

Therefore, the sum of the first 10 positive even numbers is correctly represented by the conjecture [tex]\(10 \times 11\)[/tex].

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