Question

practice make a conjecture for each scenario. show your work.

1. the sum of the first 100 positive even numbers

1. the product of two odd numbers

Explanation:

Step1: Identify the even number sequence

The first 100 positive even numbers are $2, 4, 6, ..., 200$. This is an arithmetic sequence with first term $a_1=2$, last term $a_{100}=200$, and number of terms $n=100$.

Step2: Apply arithmetic series formula

The sum of an arithmetic series is $S_n = \frac{n(a_1+a_n)}{2}$.
Substitute values: $S_{100} = \frac{100(2+200)}{2}$

Step3: Simplify the expression

First calculate inside the parentheses: $2+200=202$
Then compute: $\frac{100 \times 202}{2} = 50 \times 202 = 10100$

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Step1: Define odd numbers algebraically

Any odd number can be written as $2k+1$ and $2m+1$, where $k,m$ are integers.

Step2: Calculate the product

Multiply the two expressions: $(2k+1)(2m+1)$
Expand: $4km + 2k + 2m + 1 = 2(2km + k + m) + 1$

Step3: Classify the result

Let $p=2km + k + m$, where $p$ is an integer. The product becomes $2p+1$, which fits the form of an odd number. Test with examples: $3 \times 5=15$ (odd), $7 \times 9=63$ (odd).

Answer:

1. The sum of the first 100 positive even numbers is 10100
2. The product of two odd numbers is an odd number