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Now that we have established the core formulas for Arithmetic Progressions, it is time to put our algebraic skills to the test with Exercise 8.2! [span_2](start_span)[span_2](end_span)
In these problems, we will calculate specific terms far down the line, determine the exact position of given numbers, and seamlessly switch between explicit and recursive mathematical rules. Let's solve!
In these problems, we will calculate specific terms far down the line, determine the exact position of given numbers, and seamlessly switch between explicit and recursive mathematical rules. Let's solve!
1 Finding Specific Terms
2 Finding Positions & Validating Terms
3 Translating Explicit to Recursive Rules
Select the correct option:
If an AP has a first term \(a=3\) and a common difference \(d=5\), what is its 10th term (\(t_{10}\))? [span_15](start_span)[span_15](end_span)
Concept:
1
Use the explicit formula \(t_n = a + (n-1)d\) to instantly calculate any term. [span_16](start_span)[span_16](end_span)
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Hint:
1
Set \(n=10\). The formula becomes \(3 + (9)5\).
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Mistakes to Avoid:
!
Forgetting the \(-1\): If you calculate \(3 + 10(5) = 53\), you have found the 11th term, not the 10th!
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Solution:
\(t_{10} = a + 9d\)
\(t_{10} = 3 + 9(5) = 3 + 45 = 48\). [span_17](start_span)[span_17](end_span)
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Select the correct option:
In the AP: \(21, 18, 15 \dots\), what is the value of the common difference \(d\)? [span_18](start_span)[span_18](end_span)
Concept:
1
Calculate \(d\) by subtracting the first term from the second term: \(d = t_2 - t_1\). [span_19](start_span)[span_19](end_span)
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Hint:
1
What is \(18 - 21\)?
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Mistakes to Avoid:
!
Dropping the negative sign: Just looking at the gap "3" and writing \(d=3\) will completely break your calculations. The sequence is decreasing!
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Solution:
\(d = t_2 - t_1 = 18 - 21 = -3\). [span_20](start_span)[span_20](end_span)
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Select the correct option:
To logically prove if a number (like 0) is a valid term in an AP, you must solve the equation \(t_n = 0\). What must be true about the solved value of \(n\)? [span_21](start_span)[span_21](end_span)
Concept:
1
The variable \(n\) strictly dictates the index or position of the sequence. Positions can only be counting numbers. [span_22](start_span)[span_22](end_span)
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Hint:
1
Can someone stand in the 4.5th place in a line?
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Mistakes to Avoid:
!
Confusing the term value with position: The term itself can be zero, negative, or a fraction. But its position (\(n\)) must be a positive integer.
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Solution:
As seen when checking if 0 is a term, solving the equation yields \(n = 8\). Because 8 is a natural integer, 0 is definitively the 8th term. [span_23](start_span)[span_23](end_span)
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Select the correct option:
What is the correct recursive rule for the AP: \(11, 8, 5, 2 \dots\)? [span_24](start_span)[span_24](end_span)
Concept:
1
A recursive rule requires two absolute components: the base case (\(t_1\)) and the relationship between consecutive terms. [span_25](start_span)[span_25](end_span)
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Hint:
1
Look for the option that clearly states the first term is 11, AND shows the subtraction of 3 from the previous term.
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Mistakes to Avoid:
!
Choosing incomplete rules: Option B tells you to subtract 3, but without telling you where to start, it is mathematically useless!
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Solution:
The first term \(a = 11\). The difference is \(-3\).
Therefore, the full recursive rule is \(t_1 = 11\) and \(t_n = t_{n-1} - 3\). [span_26](start_span)[span_26](end_span)
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Select the correct option:
If you are told that the 8th term of an Arithmetic Progression is exactly 0, which algebraic equation must be true?
Concept:
1
Using \(t_n = a + (n-1)d\), any specific term gives us a linear equation tying \(a\) and \(d\) together. [span_27](start_span)[span_27](end_span)
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Hint:
1
Substitute \(n=8\) into \(a + (n-1)d\).
2
Set the result equal to 0.
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Mistakes to Avoid:
!
Using \(nd\): Assuming the 8th term is \(a + 8d\) is a fatal error that will ruin simultaneous equation problems! Always use \((n-1)\).
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Solution:
\(t_n = a + (n-1)d\)
\(t_8 = a + (8-1)d\)
\(a + 7d = 0\)
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