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Now that we understand what sequences are and how explicit rules act as a mathematical "time machine" to jump to any position, let us put the concept into practice!
In Exercise 8.1, we will master the art of generating entire sequences from complex algebraic formulas and hunting down specific terms using substitution. [span_1](start_span)[span_1](end_span)
In Exercise 8.1, we will master the art of generating entire sequences from complex algebraic formulas and hunting down specific terms using substitution. [span_1](start_span)[span_1](end_span)
1 Generating Explicit Sequences
2 Calculating Values & Checking Terms
Select the correct option:
If a sequence is defined by \(t_n = 4n + 1\), what is the value of \(t_3\)?
Concept:
1
To find a specific term using an explicit rule, substitute the required position number for the variable \(n\).
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Hint:
1
Set \(n = 3\). Calculate \(4 \times 3 + 1\).
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Mistakes to Avoid:
!
Order of Operations error: If you calculate \(4 \times (3+1)\), you get 16. Multiplication must happen before addition!
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Solution:
Given: \(t_n = 4n + 1\)
For the 3rd term, substitute \(n = 3\):
\(t_3 = 4(3) + 1\)
\(t_3 = 12 + 1 = 13\)
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Select the correct option:
If a sequence generates fractions like \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4} \dots\), which explicit formula accurately describes it? [span_10](start_span)[span_10](end_span)
Concept:
1
Explicit formulas can generate fractions. You must verify that the formula works perfectly starting from \(n=1\). [span_11](start_span)[span_11](end_span)
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Hint:
1
Since the first term is \(\frac{1}{2}\), substituting \(n=1\) into the correct formula must equal \(\frac{1}{2}\).
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Mistakes to Avoid:
!
Assuming \(1/n\) works: While the sequence looks like \(1/n\), you must remember that sequences start at \(n=1\). If the first term is \(1/2\), the denominator is \(n+1\).
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Solution:
Test formula B: \(t_n = \frac{1}{n+1}\)
\(t_1 = \frac{1}{1+1} = \frac{1}{2}\)
\(t_2 = \frac{1}{2+1} = \frac{1}{3}\)
This perfectly matches the sequence. [span_12](start_span)[span_12](end_span)
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Select the correct option:
Which term of the sequence given by \(t_n = 7n - 2\) is 54?
Concept:
1
To find which term a number represents, set the formula equal to the number and solve algebraically for \(n\).
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Hint:
1
Set \(7n - 2 = 54\).
2
Add 2 to both sides.
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Mistakes to Avoid:
!
Subtracting instead of adding: When moving the \(-2\) across the equals sign, it becomes \(+2\). \(54 + 2 = 56\). Do not calculate \(54 - 2\).
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Solution:
Let \(t_n = 54\)
\(7n - 2 = 54\)
\(7n = 54 + 2\)
\(7n = 56\)
\(n = \frac{56}{7} = 8\)
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Select the correct option:
Why might a specific number NOT belong to a sequence given by \(t_n = 2n + 5\)?
Concept:
1
Because \(n\) represents a physical position (1st, 2nd, 3rd...), it MUST be a positive natural integer. [span_13](start_span)[span_13](end_span)
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Hint:
1
Can you have a 3.5th term? Can you have a -2nd term?
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Mistakes to Avoid:
!
Accepting any value for n: In algebra, \(x\) can be a decimal or negative. In sequences, the index \(n\) is strictly a counting number.
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Solution:
If solving an equation for \(n\) yields a fraction, a decimal, or a negative number, the given value cannot be a term of the sequence. Therefore, both B and C are correct reasons.
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Select the correct option:
What is the explicit rule for the sequence of even natural numbers \(2, 4, 6, 8, 10 \dots\)?
Concept:
1
A sequence that increases by a constant multiplier (e.g., adding 2 each time) will have that constant multiplied by \(n\).
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Hint:
1
Substitute \(n=1\). Which formula gives 2?
2
Substitute \(n=2\). Which formula gives 4?
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Mistakes to Avoid:
!
Confusing with \(n+2\): \(n+2\) means you add 2 to the position number. For \(n=1\), it is \(1+2=3\), which is not an even number sequence starting at 2.
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Solution:
Testing \(t_n = 2n\):
\(t_1 = 2(1) = 2\)
\(t_2 = 2(2) = 4\)
\(t_3 = 2(3) = 6\)
This perfectly generates the sequence of even numbers.
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★ Complex Sequence Algebra
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