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We have seen how to find the common difference in simple integer sequences. Now, let's level up!
Arithmetic Progressions can be built with fractions and decimals too. More importantly, we can use the explicit formula \(t_n = a + (n-1)d\) and recursive formulas \(t_n = t_{n-1} + d\) to calculate real-world scenarios, like the exact fare of a long taxi ride without counting every single kilometer! [span_2](start_span)[span_2](end_span)[span_3](start_span)[span_3](end_span)
Arithmetic Progressions can be built with fractions and decimals too. More importantly, we can use the explicit formula \(t_n = a + (n-1)d\) and recursive formulas \(t_n = t_{n-1} + d\) to calculate real-world scenarios, like the exact fare of a long taxi ride without counting every single kilometer! [span_2](start_span)[span_2](end_span)[span_3](start_span)[span_3](end_span)
1 Fractional AP Rules
2 Decimal AP Rules
3 Real-World AP: Taxi Fare Math
Select the correct option:
What is the correct format for the recursive rule of a standard Arithmetic Progression? [span_23](start_span)[span_23](end_span)
Concept:
1
A recursive rule for an AP must define the starting value (\(t_1\)) and state that the next term is found by adding the common difference (\(d\)) to the current term (\(t_{n-1}\)). [span_24](start_span)[span_24](end_span)
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Hint:
1
Look for the option that uses \(t_{n-1}\) AND addition.
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Mistakes to Avoid:
!
Confusing Explicit vs Recursive: Option A is a perfectly valid AP formula, but it is explicitly defined, not recursively. Pay attention to the terminology requested! [span_25](start_span)[span_25](end_span)[span_26](start_span)[span_26](end_span)
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Solution:
As stated in the textbook, the recursive rule for any Arithmetic Progression is \(t_1 = a\), \(t_n = t_{n-1} + d\) for \(n \ge 2\). [span_27](start_span)[span_27](end_span)
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Select the correct option:
If the first term of an AP is \(a = \frac{1}{2}\) and the common difference is \(d = 2\), what is the value of the 3rd term (\(t_3\))?
Concept:
1
You can find a specific term by manually adding \(d\) successively, or by using the explicit formula \(t_n = a + (n-1)d\).
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Hint:
1
Using the formula: \(t_3 = \frac{1}{2} + (3-1)2\).
2
\(t_3 = \frac{1}{2} + 4\). Make common denominators to add!
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Mistakes to Avoid:
!
Adding fractions incorrectly: \(1/2 + 2\) is not \(3/2\). You must convert 2 into halves (\(4/2\)) before adding: \(1/2 + 4/2 = 5/2\).
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Solution:
Given: \(a = \frac{1}{2}\) and \(d = 2\).
\(t_3 = a + 2d\)
\(t_3 = \frac{1}{2} + 2(2)\)
\(t_3 = \frac{1}{2} + 4\)
\(t_3 = \frac{1}{2} + \frac{8}{2} = \frac{9}{2}\)
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Select the correct option:
A gym charges a fixed membership joining fee of ₹100, plus ₹20 per day. What is the total cost after 5 days? [span_28](start_span)[span_28](end_span)
Concept:
1
A real-world fixed cost plus a recurring rate is a linear model (AP). [span_29](start_span)[span_29](end_span)
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Hint:
1
Day 1 Cost = 100 + 20 = 120. So \(a = 120\).
2
Find \(t_5\) using \(d = 20\).
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Mistakes to Avoid:
!
Treating the fixed fee as \(a\): The fixed fee of 100 is for Day 0. If you set \(a = 100\) and calculate \(t_5 = 100 + (5-1)20\), you get 180, which is wrong! Day 1 is 120, so the formula is \(120 + 4(20) = 200\). [span_30](start_span)[span_30](end_span)
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Solution:
Total Cost = Fixed Fee + (Rate \(\times\) Number of Days)
Cost = \(100 + 20(5)\)
Cost = \(100 + 100 = 200\)
This matches the AP logic where \(a = 120\) (Day 1) and \(d = 20\): \(t_5 = 120 + 4(20) = 200\). [span_31](start_span)[span_31](end_span)
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Select the correct option:
What is the explicit formula (\(t_n\)) for an Arithmetic Progression that starts at \(a = 1.5\) and has a common difference of \(d = 2.0\)? [span_32](start_span)[span_32](end_span)
Concept:
1
Always use the master formula \(t_n = a + (n-1)d\) and simplify algebraically. [span_33](start_span)[span_33](end_span)
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Hint:
1
Substitute: \(t_n = 1.5 + (n-1)2.0\)
2
Expand: \(t_n = 1.5 + 2n - 2.0\)
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Mistakes to Avoid:
!
Assuming \(a\) is the y-intercept: In the linear formula \(t_n = dn + c\), the constant \(c\) is NOT the first term \(a\). It is actually \(a - d\). So do not just guess Option A!
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Solution:
Given: \(a = 1.5\) and \(d = 2.0\).
\(t_n = 1.5 + (n-1)2\)
\(t_n = 1.5 + 2n - 2\)
\(t_n = 2n - 0.5\)
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Select the correct option:
Identify the common difference in the AP: \(1.5, 3.5, 5.5, 7.5 \dots\) [span_34](start_span)[span_34](end_span)
Concept:
1
The common difference (\(d\)) is calculated by subtracting any term from the term immediately following it (\(t_2 - t_1\)). [span_35](start_span)[span_35](end_span)
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Hint:
1
Calculate \(3.5 - 1.5\).
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Mistakes to Avoid:
!
Confusing \(a\) and \(d\): Often students see the first number and instantly select it. Always read carefully if the question asks for the "first term" or the "common difference"!
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Solution:
Given AP: \(1.5, 3.5, 5.5 \dots\)
\(d = t_2 - t_1\)
\(d = 3.5 - 1.5 = 2.0\)
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