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Up until now, we have looked at sequences with varying rules. However, there is a special, highly predictable family of sequences called Arithmetic Progressions (APs). [span_3](start_span)[span_3](end_span)
In an AP, the difference between any two consecutive terms is always exactly the same. We call the first term \(a\), and the constant jump between terms the common difference \((d)\). Because this growth is perfectly constant, if you plot an AP on a graph, it forms a perfect straight line! [span_4](start_span)[span_4](end_span)[span_5](start_span)[span_5](end_span)
In an AP, the difference between any two consecutive terms is always exactly the same. We call the first term \(a\), and the constant jump between terms the common difference \((d)\). Because this growth is perfectly constant, if you plot an AP on a graph, it forms a perfect straight line! [span_4](start_span)[span_4](end_span)[span_5](start_span)[span_5](end_span)
1 Modelling Growing Patterns
2 Identifying APs in Previous Concepts
3 Identifying Negative APs
Select the correct option:
Which of the following defines the common difference (\(d\)) of an Arithmetic Progression (AP)? [span_16](start_span)[span_16](end_span)
Concept:
1
An arithmetic progression (AP) can be described as \(a, a+d, a+2d, a+3d \dots\) where \(d\) is the common difference. [span_17](start_span)[span_17](end_span)
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Hint:
1
If a sequence goes \(10, 15, 20\), you are adding \(5\) each time. What do we call the \(5\)? [span_18](start_span)[span_18](end_span)
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Mistakes to Avoid:
!
Confusing subtraction order: Never calculate \(d = t_1 - t_2\). It must always be calculated as \(d = t_2 - t_1\) or \(d = t_3 - t_2\).
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Solution:
As defined in the textbook, the common difference \(d\) is the constant value added to any term to generate the consecutive term. [span_19](start_span)[span_19](end_span)
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Select the correct option:
When you plot the terms of an Arithmetic Progression on a graph (where x is the position number and y is the term value), what geometric shape emerges? [span_20](start_span)[span_20](end_span)
Concept:
1
Because an AP formula \(t_n = an + b\) is a linear equation (degree 1), its graph will always be a line. [span_21](start_span)[span_21](end_span)
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Hint:
1
Think of the points \((1, 1), (2, 5), (3, 9)\). If you connect them, what do they draw? [span_22](start_span)[span_22](end_span)
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Mistakes to Avoid:
!
Assuming all patterns curve: Many natural patterns curve (like populations or areas), but basic APs are strictly linear.
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Solution:
As illustrated in Figure 8.4 of the textbook, when we plot the ordered pairs emerging from an AP, we observe that they lie on a perfectly straight line! [span_23](start_span)[span_23](end_span)
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Select the correct option:
Which of the following sequences is an Arithmetic Progression? [span_24](start_span)[span_24](end_span)
Concept:
1
A sequence is an AP if and only if \(t_2 - t_1 = t_3 - t_2\). The difference must never change. [span_25](start_span)[span_25](end_span)
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Hint:
1
Test Option B: \(-2 - (-5) = ?\) and \(1 - (-2) = ?\)
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Mistakes to Avoid:
!
Confusing multiples with addition: Option D looks like a pattern, but it is geometric (multiplying by 2), not arithmetic!
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Solution:
Checking Option B: \(-5, -2, 1, 4\)
\(-2 - (-5) = 3\)
\(1 - (-2) = 3\)
\(4 - 1 = 3\)
Since the difference is a constant \(3\), it is a valid Arithmetic Progression. [span_26](start_span)[span_26](end_span)
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Select the correct option:
In the AP: \(11, 7, 3, -1, -5 \dots\), what are the values of the first term (\(a\)) and the common difference (\(d\))? [span_27](start_span)[span_27](end_span)
Concept:
1
A sequence that decreases means the common difference \(d\) is a negative number. [span_28](start_span)[span_28](end_span)
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Hint:
1
Calculate \(d = t_2 - t_1\).
2
\(d = 7 - 11 = ?\)
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Mistakes to Avoid:
!
Assuming \(d\) is always positive: If you just look at the gap (4) and write \(d=4\), you are generating the sequence \(11, 15, 19\). You must respect the sign! [span_29](start_span)[span_29](end_span)
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Solution:
The first term \(a\) is strictly the first number in the list: \(a = 11\).
The common difference \(d\) is calculated as \(t_2 - t_1\):
\(d = 7 - 11 = -4\)
As noted in the textbook, the numbers decrease by 4, making \(d = -4\). [span_30](start_span)[span_30](end_span)
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Select the correct option:
In the general explicit formula for an Arithmetic Progression, \(t_n = a + (n-1)d\), why do we multiply \(d\) by \((n-1)\) instead of just \(n\)? [span_31](start_span)[span_31](end_span)
Concept:
1
The formula \(t_n = a + (n-1)d\) represents starting at the base point (\(a\)) and adding the gap (\(d\)) repeatedly. [span_32](start_span)[span_32](end_span)
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Hint:
1
To find \(t_1\), if we used \(nd\), we would get \(a + 1d\). But the first term should just be \(a\)!
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Mistakes to Avoid:
!
Using \(a + nd\): This is a very common mistake. If you use \(a + nd\), you are actually calculating the term after \(n\).
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Solution:
By logic, the 1st term requires 0 jumps of \(d\).
The 2nd term requires 1 jump of \(d\).
The 3rd term requires 2 jumps of \(d\).
Therefore, the \(n^{th}\) term requires \((n-1)\) jumps of \(d\). [span_33](start_span)[span_33](end_span)
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