Geometric Progressions are not only found in abstract math; they govern physical motion, such as a bouncing ball.
When a ball is dropped, every time it hits the floor, it loses a fixed percentage of its height, rebounding to a fraction of its previous peak. This forms a decreasing GP. Furthermore, when plotted on a graph, exponential sequences curve sharply—unlike the straight lines of APs.
When a ball is dropped, every time it hits the floor, it loses a fixed percentage of its height, rebounding to a fraction of its previous peak. This forms a decreasing GP. Furthermore, when plotted on a graph, exponential sequences curve sharply—unlike the straight lines of APs.
1 Bouncing Ball Physics Model
2 Analyzing GP Graphical Growth
Select the correct option:
A ball is dropped from a height of 80 meters and rebounds to half of its height on every bounce. What is the height of the ball after the 3rd bounce?
Concept:
1
A bouncing ball creates a shrinking GP where each term is multiplied by the fractional rebound ratio \(r\).
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Hint:
1
Multiply 80 by \(\frac{1}{2}\) three consecutive times.
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Mistakes to Avoid:
!
Dividing by 3 instead of halving 3 times: Halving 3 times means dividing by \(2^3 = 8\). Do not divide 80 by 3!
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Solution:
1st bounce: \(80 \times \frac{1}{2} = 40\)
2nd bounce: \(40 \times \frac{1}{2} = 20\)
3rd bounce: \(20 \times \frac{1}{2} = 10\text{ m}\).
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Select the correct option:
When plotting an Arithmetic Progression versus a Geometric Progression on a standard grid, how do their graphical representations differ?
Concept:
1
Linear equations (addition) draw lines. Exponential equations (multiplication/powers) draw curves.
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Hint:
1
Recall Figure 8.4 (AP straight line) versus GP exponential escalation.
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Mistakes to Avoid:
!
Treating exponential growth as linear: Multipliers compound rapidly. Connecting GP points with a ruler will result in a severely inaccurate graph!
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Solution:
APs produce a constant slope resulting in a straight line, whereas GPs compound rapidly, producing a curved trajectory on a coordinate grid.
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Select the correct option:
If a ball rebounds to \(\frac{2}{3}\) of its height after each drop, what is the common ratio (\(r\)) of the bounce height sequence?
Concept:
1
In physical drop-and-bounce models, the fraction of the height retained is the common ratio \(r\) of the GP.
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Hint:
1
Each term is multiplied by \(\frac{2}{3}\) to find the next term.
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Mistakes to Avoid:
!
Inverting the fraction: Do not write \(\frac{3}{2}\). A ratio of \(\frac{3}{2}\) (1.5) means the ball would bounce higher than where it was dropped, defying gravity!
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Solution:
The fraction of height retained after each bounce is directly the common ratio: \(r = \frac{2}{3}\).
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Select the correct option:
What type of mathematical growth or decay does a Geometric Progression represent on a coordinate plane?
Concept:
1
Because explicit GP formulas contain exponents (like \(3^n\)), they belong to the family of exponential functions.
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Hint:
1
Think of virus replication or money compounding with interest. They grow exponentially!
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Mistakes to Avoid:
!
Confusing linear and exponential: Linear adds equal chunks; exponential multiplies by factors.
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Solution:
Geometric progressions feature variables in exponents, classifying them strictly as exponential growth or decay functions.
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Select the correct option:
If a bouncing ball's heights form the GP: \(64, 32, 16, 8 \dots\), what will be the height of the ball after the 5th bounce?
Concept:
1
Continue halving the sequence step-by-step until you reach the 5th term.
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Hint:
1
The 4th term is given as 8. Halve 8 to get the 5th term.
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Mistakes to Avoid:
!
Stopping at 4th bounce: 8 is already listed as the 4th bounce. You must take one more step to find the 5th!
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Solution:
Sequence: \(64, 32, 16, 8\) (4th bounce)
5th bounce = \(8 \times \frac{1}{2} = 4\text{ m}\).
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