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To describe mathematical rules for sequences, we use notations like \(t_n, s_n,\) or \(u_n\) to represent the term at the \(n^{th}\) position. [span_2](start_span)[span_2](end_span)
An explicit rule uses the position number \(n\) to calculate the term's value directly. For example, if the formula is \(u_n = 2n - 1\), you can find the 100th term instantly by just substituting \(n = 100\), without ever needing to know the 99th term! [span_3](start_span)[span_3](end_span)[span_4](start_span)[span_4](end_span) Let's master explicit formulas.
An explicit rule uses the position number \(n\) to calculate the term's value directly. For example, if the formula is \(u_n = 2n - 1\), you can find the 100th term instantly by just substituting \(n = 100\), without ever needing to know the 99th term! [span_3](start_span)[span_3](end_span)[span_4](start_span)[span_4](end_span) Let's master explicit formulas.
1 Evaluating Explicit Rules
2 Checking if a Number is a Term
Select the correct option:
What is the primary advantage of having an explicit formula for a sequence? [span_13](start_span)[span_13](end_span)
Concept:
1
An explicit formula (like \(t_n = 2n + 1\)) is an independent equation. [span_14](start_span)[span_14](end_span)[span_15](start_span)[span_15](end_span)
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Hint:
1
If I want the 1,000,000th term, do I need to calculate the 999,999th term first? [span_16](start_span)[span_16](end_span)
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Mistakes to Avoid:
!
Confusing Explicit with Recursive: Option A describes a recursive formula, where you must know the previous step to take the next one.
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Solution:
As stated in the textbook, using an explicit formula, we can find the value of a term without having to know the value of previous terms. [span_17](start_span)[span_17](end_span)
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Select the correct option:
If a sequence is defined by the explicit rule \(t_n = 3n + 2\), what is the value of the 5th term?
Concept:
1
To find a specific term using an explicit rule, substitute the required position number for the variable \(n\). [span_18](start_span)[span_18](end_span)
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Hint:
1
Set \(n = 5\). Calculate \(3 \times 5 + 2\).
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Mistakes to Avoid:
!
Order of Operations error: If you calculate \(3 \times (5+2)\), you get 21. Multiplication must happen before addition!
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Solution:
Given: \(t_n = 3n + 2\)
For the 5th term, substitute \(n = 5\):
\(t_5 = 3(5) + 2\)
\(t_5 = 15 + 2 = 17\)
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Select the correct option:
In sequence notation, how is the first term of a sequence commonly represented? [span_19](start_span)[span_19](end_span)
Concept:
1
In sequence notation, the letter (like \(t, s, u\)) represents the sequence, and the small subscript number represents the position. [span_20](start_span)[span_20](end_span)
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Hint:
1
The first position uses the number 1.
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Mistakes to Avoid:
!
Starting at zero: Unlike some computer programming languages that start counting at 0, mathematical sequences almost always start at position 1. [span_21](start_span)[span_21](end_span)
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Solution:
As detailed in the textbook notation, we use \(t_1\) to represent the first term of a sequence, \(t_2\) for the second, and so on. [span_22](start_span)[span_22](end_span)
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Select the correct option:
When solving an equation to check if a number belongs to a sequence, why must the resulting value for n be a natural number? [span_23](start_span)[span_23](end_span)
Concept:
1
In sequence notation, \(n\) refers strictly to the index or position of a term. [span_24](start_span)[span_24](end_span)[span_25](start_span)[span_25](end_span)
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Hint:
1
Think of people standing in a line. You can have the 1st person, 2nd person, and 3rd person. Can you have the 2.5th person? [span_26](start_span)[span_26](end_span)
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Mistakes to Avoid:
!
Confusing the term value with the position: The term value can be anything (fraction, decimal, negative). But the position number (\(n\)) must always be a clean, positive integer (1, 2, 3...). [span_27](start_span)[span_27](end_span)
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Solution:
As noted in the textbook, \(n\) must be a non-negative natural integer because it represents the position in the sequence (1st, 2nd, 3rd). A position cannot be a fraction or a negative number. [span_28](start_span)[span_28](end_span)[span_29](start_span)[span_29](end_span)
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Select the correct option:
If a sequence has the explicit rule \(u_n = n^2 + 1\), what is the value of the 4th term?
Concept:
1
Explicit formulas can contain exponents. Substitute \(n\) carefully and follow order of operations (exponents before addition).
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Hint:
1
Set \(n = 4\).
2
Calculate \(4^2\), then add 1.
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Mistakes to Avoid:
!
Adding before squaring: If you calculate \((4+1)^2\), you will get \(5^2 = 25\), which is wrong.
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Solution:
Given: \(u_n = n^2 + 1\)
For the 4th term, substitute \(n = 4\):
\(u_4 = (4)^2 + 1\)
\(u_4 = 16 + 1 = 17\)
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★ Complex Explicit Formulas
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