IDNLearn.com: Where your questions meet expert advice and community support. Join our interactive Q&A community and get reliable, detailed answers from experienced professionals across a variety of topics.
Prove: The square of a number that is one more than a multiple of 3 is also one more than a multiple of 3.
[tex]
\begin{aligned}
(3n + 1)^2 & = 9n^2 + 6n + 1 \\
& = 3(3n^2 + 2n) + 1 \\
& = 1 \text{ more than a multiple of } 3
\end{aligned}
[/tex]
Certainly! Let's go through the problem step-by-step to prove that the square of a number that is one more than a multiple of 3 is also one more than a multiple of 3.
We start with the expression [tex]\((3n + 1)^2\)[/tex].
1. Expand the expression: [tex]\[
(3n + 1)^2 = (3n + 1)(3n + 1)
\][/tex] Use the distributive property (also known as FOIL in this specific case): [tex]\[
(3n + 1)(3n + 1) = (3n)(3n) + (3n)(1) + (1)(3n) + (1)(1)
\][/tex]
3. Add all the terms together: [tex]\[
9n^2 + 3n + 3n + 1
\][/tex] Combine like terms: [tex]\[
9n^2 + 6n + 1
\][/tex]
We now have the expanded expression: [tex]\[
(3n + 1)^2 = 9n^2 + 6n + 1
\][/tex]
4. Analyze the result: Notice that [tex]\(9n^2 + 6n\)[/tex] is a multiple of 3 because both [tex]\(9n^2\)[/tex] and [tex]\(6n\)[/tex] are individually multiples of 3.
Indeed, 9 and 6 are multiples of 3: [tex]\[
9n^2 = 3 \cdot 3n^2 \quad \text{and} \quad 6n = 3 \cdot 2n
\][/tex]
5. Express [tex]\(9n^2 + 6n + 1\)[/tex] as one more than a multiple of 3: Factor the expression to show it clearly: [tex]\[
9n^2 + 6n + 1 = 3(3n^2 + 2n) + 1
\][/tex]
This shows that the term [tex]\(9n^2 + 6n\)[/tex] is a multiple of 3, and adding 1 makes the whole expression [tex]\(9n^2 + 6n + 1\)[/tex] one more than a multiple of 3.
Therefore, [tex]\((3n + 1)^2\)[/tex] is indeed one more than a multiple of 3.
In conclusion: [tex]\[
(3n + 1)^2 = 9n^2 + 6n + 1
\][/tex] is clearly shown to be: [tex]\[
1 \text{ more than a multiple of } 3
\][/tex]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! IDNLearn.com is your reliable source for answers. We appreciate your visit and look forward to assisting you again soon.
