Congratulations! You have reached the very end of Chapter 8: Sequences and Progressions (Page 196).
We began by discovering simple patterns in nature and numbers. We then learned how to build "time machines" (explicit formulas) to jump to any point in the future, and "stepping stones" (recursive formulas) to walk there step-by-step. We mastered the linear paths of Arithmetic Progressions (APs) and the explosive, curved growth of Geometric Progressions (GPs). Let's lock in everything you've learned.
We began by discovering simple patterns in nature and numbers. We then learned how to build "time machines" (explicit formulas) to jump to any point in the future, and "stepping stones" (recursive formulas) to walk there step-by-step. We mastered the linear paths of Arithmetic Progressions (APs) and the explosive, curved growth of Geometric Progressions (GPs). Let's lock in everything you've learned.
1 The Master Cheat Sheet
2 Scenario Sorting
Select the correct option:
Which of the following is the correct explicit formula for an Arithmetic Progression?
Concept:
1
AP relies on constant addition, requiring the variable \(d\).
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Hint:
1
Look for the formula featuring addition (+) and the common difference (\(d\)).
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Mistakes to Avoid:
!
Confusing Explicit and Sum formulas: Option B is a sum formula. Option C is the term formula.
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Solution:
The explicit rule for an AP is \(t_n = a + (n-1)d\).
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Select the correct option:
If you graph a progression and the points curve steeply upwards, what type of mathematical growth does this represent?
Concept:
1
A straight line = Linear = AP.
2
A steep curve = Exponential = GP.
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Hint:
1
Recall the visual difference between the graphs in Chapter 8. Which one featured the exponent \(n\)?
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Mistakes to Avoid:
!
Assuming all lines curve eventually: An AP graph will stay perfectly straight into infinity.
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Solution:
A curve indicates that the gap between terms is multiplying, which is the defining trait of exponential growth found in Geometric Progressions.
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Select the correct option:
Which of the following is a pure recursive definition of an Arithmetic Progression?
Concept:
1
Recursive formulas rely on the previous term (\(t_{n-1}\)).
2
Arithmetic progressions rely on addition (\(+d\)).
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Hint:
1
Find the option that combines \(t_{n-1}\) with addition.
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Mistakes to Avoid:
!
Ignoring the base case: A true recursive formula MUST state the first term (\(t_1\)). If an option just says \(t_n = t_{n-1} + d\), it is incomplete.
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Solution:
Option C is the only complete recursive definition of an AP.
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Select the correct option:
In both the AP and GP formulas, what physical concept does the variable \(n\) represent?
Concept:
1
The variable \(n\) dictates where you are in the sequence. It is the counter.
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Hint:
1
If a question asks "Which term is 100?", you are solving for \(n\).
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Mistakes to Avoid:
!
Confusing \(n\) with \(t_n\): \(n\) is the location (e.g., house number 4). \(t_n\) is the value at that location (e.g., the family living inside house number 4).
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Solution:
\(n\) stands for the position index, which is why it must be a positive natural integer (you cannot be in the 3.5th position).
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Select the correct option:
What is the sum of the first 100 natural numbers? (Use the formula \(S_n = \frac{n(n+1)}{2}\))
Concept:
1
Substitute \(n=100\) directly into the summation formula.
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Hint:
1
\(\frac{100 \times 101}{2}\).
2
Simplify \(\frac{100}{2}\) to \(50\), then multiply by 101.
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Mistakes to Avoid:
!
Forgetting the denominator: Many students evaluate \(100 \times 101 = 10100\) and select that. You must halve the rectangle to find the triangle area!
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Solution:
\(S_{100} = \frac{100(101)}{2}\)
\(S_{100} = 50 \times 101 = 5050\).
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