Evaluate Term-to-Term Rules: Comparing 7n-2 vs 9n+4-2n-2 in Age Sequences

Q: How do I know which expressions are equivalent without testing every value?

Simplify each expression first! If two expressions simplify to the same form (like $ 7n - 2 $), they're equivalent. This saves time compared to testing multiple values.

Q: What does it mean when variables like x² cancel out?

When terms like $ x^2 - x^2 $ appear, they cancel to zero and disappear from the expression. The remaining terms determine the final simplified form.

Q: Why do we test with small values like n = 1, 2, 3?

Small values are easy to calculate and help us quickly check if our rule matches the given sequence. If it works for the first few terms, our rule is likely correct!

Q: Can different-looking expressions give the same sequence?

Absolutely! Expressions like $ 7n - 2 $, $ 9n + 4 - 2n - 2 $, and $ x^2 + 5n - x^2 + 2n - 2 $ all simplify to the same thing, so they produce identical sequences.

Q: What if my simplified expression doesn't match the sequence?

That means the original expression is not a valid rule for this sequence. Double-check your simplification, then test with the given values to confirm.

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

A group of mathematicians decide in advance on a term-to-term rule for a sequence.

They then find people whose ages match the rule and line them up in the following order:

Which of the following are appropriate term-to-term rules?

a. $9n+4-2n-2$

b. $x^2+5n-x^2+2n-2$

c. $7n-2$

d. $9n+4-n-6-n$

2

Step-by-step solution

To solve this problem, we will evaluate each expression by substituting small values of $n$ (1, 2, and 3) and comparing the results to the given sequence 5, 12, 19, ...

We'll examine each option:

Option a: $9n + 4 - 2n - 2$

The expression simplifies to $9n - 2n + 4 - 2 = 7n + 2$ .

Substitute $n = 1,$ then $7(1) + 2 = 9.$ This does not match 5.

Hence, this rule is unsuitable.

Option b: $x^2 + 5n - x^2 + 2n - 2$

This simplifies to $5n + 2n - 2 = 7n - 2$ .

Substitute $n = 1,$ then $7(1) - 2 = 5,$ which matches.

Substitute $n = 2,$ then $7(2) - 2 = 12,$ which matches.

Substitute $n = 3,$ then $7(3) - 2 = 19,$ which matches.

This rule is suitable.

Option c: $7n - 2$

This matches the rule used in option b.

Hence, this rule is suitable as well.

Option d: $9n + 4 - n - 6 - n$

The expression simplifies to $9n - 2n + 4 - 6 = 7n - 2$ .

This matches the results in options b and c when evaluated.

This rule is also suitable.

Therefore, the rules described in options b, c, and d generate the ages sequence correctly. All of these simplify to $7n - 2$ .

The correct answer is choices b, d, and c.

3

Final Answer

b, d, and c

Key Points to Remember

Essential concepts to master this topic

Simplification: Combine like terms by adding/subtracting coefficients of same variable
Technique: For $9n + 4 - 2n - 2$ , group terms: $(9n - 2n) + (4 - 2) = 7n + 2$
Check: Substitute n = 1, 2, 3 into simplified form and verify sequence matches ✓

Common Mistakes

Avoid these frequent errors

Not simplifying expressions before testing
Don't substitute values into unsimplified expressions like $9n + 4 - 2n - 2$ without combining like terms first = messy calculations and possible errors! This makes it harder to see if expressions are equivalent. Always simplify by combining like terms before substituting test values.

Practice Quiz

Test your knowledge with interactive questions

Are the expressions the same or not?

$$ 3+3+3+3 $$

$3\times4$

FAQ

Everything you need to know about this question

How do I know which expressions are equivalent without testing every value?

+

Simplify each expression first! If two expressions simplify to the same form (like $7n - 2$ ), they're equivalent. This saves time compared to testing multiple values.

What does it mean when variables like x² cancel out?

+

When terms like $x^2 - x^2$ appear, they cancel to zero and disappear from the expression. The remaining terms determine the final simplified form.

Why do we test with small values like n = 1, 2, 3?

+

Small values are easy to calculate and help us quickly check if our rule matches the given sequence. If it works for the first few terms, our rule is likely correct!

Can different-looking expressions give the same sequence?

+

Absolutely! Expressions like $7n - 2$ , $9n + 4 - 2n - 2$ , and $x^2 + 5n - x^2 + 2n - 2$ all simplify to the same thing, so they produce identical sequences.

What if my simplified expression doesn't match the sequence?

+

That means the original expression is not a valid rule for this sequence. Double-check your simplification, then test with the given values to confirm.

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Evaluate Term-to-Term Rules: Comparing 7n-2 vs 9n+4-2n-2 in Age Sequences

Algebraic Expressions with Equivalent Simplifications

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Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I know which expressions are equivalent without testing every value?

What does it mean when variables like x² cancel out?

Why do we test with small values like n = 1, 2, 3?

Can different-looking expressions give the same sequence?

What if my simplified expression doesn't match the sequence?

🌟 Unlock Your Math Potential

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