Arithmetic Progressions are not just abstract numbers on a page; they govern real-world systems built on constant growth or decline.
Whether it is tracking the exact amount of money saved in a bank account every month, or calculating the declining number of seats in a stadium section row by row, you can map real life directly onto the formula \(t_n = a + (n-1)d\). Let's finish Exercise 8.2 by decoding these word problems!
Whether it is tracking the exact amount of money saved in a bank account every month, or calculating the declining number of seats in a stadium section row by row, you can map real life directly onto the formula \(t_n = a + (n-1)d\). Let's finish Exercise 8.2 by decoding these word problems!
1 Financial Growth (Savings)
2 Decreasing Physical Patterns
3 Salary Increments
Select the correct option:
In a word problem where a person's starting salary is ₹8000 and it increases by a fixed ₹500 every year, what mathematical variable does the ₹500 represent?
Concept:
1
In real-world linear models, the initial value is \(a\), and the constant rate of change (increment or decrement) is \(d\).
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Hint:
1
If Year 1 is 8000, and Year 2 is 8500, what is the gap between them?
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Mistakes to Avoid:
!
Confusing variables: Always explicitly define \(a\), \(d\), \(n\), and \(t_n\) before solving a word problem to avoid plugging numbers into the wrong parts of the formula.
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Solution:
The value that is added consistently every single step (the fixed increment) is the common difference, denoted by \(d\).
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Select the correct option:
A taxi fare is ₹15 for the first kilometer, and ₹8 for each additional kilometer. What is the total fare for a journey of 4 km?
Concept:
1
This is an AP where \(a=15\), \(d=8\), and we want to find the term at position \(n=4\).
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Hint:
1
Calculate \(15 + (4-1)8\).
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Mistakes to Avoid:
!
Multiplying by \(n\): If you calculate \(15 + 4(8) = 47\), you are paying the extra rate for 4 kilometers on top of the base km. You only drove 3 additional kilometers!
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Solution:
\(a = 15\), \(d = 8\), \(n = 4\)
\(t_n = a + (n-1)d\)
\(t_4 = 15 + (4-1)8\)
\(t_4 = 15 + 3(8) = 15 + 24 = 39\)
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Select the correct option:
If a word problem states "the water level in a tank decreases by 5 liters every minute", what is the value of the common difference (\(d\))?
Concept:
1
Words like "loss", "decrease", "depreciation", or "draining" at a constant rate always signify a negative common difference (\(-d\)).
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Hint:
1
If you start with 100 liters, min 1 is 100, min 2 is 95. What is \(95 - 100\)?
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Mistakes to Avoid:
!
Ignoring context: In math, the word "decreases" is an explicit instruction to append a negative sign to your rate variable.
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Solution:
Since the rate is a constant subtraction of 5 liters per step, the common difference \(d = -5\).
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Select the correct option:
Ramkali saved ₹5 in the first week of a year and increased her weekly savings by ₹1.75. What will her savings be in the 10th week?
Concept:
1
Use the explicit formula \(t_n = a + (n-1)d\) with decimals.
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Hint:
1
\(a = 5\), \(d = 1.75\), \(n = 10\).
2
Calculate \(5 + 9(1.75)\).
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Mistakes to Avoid:
!
Multiplying by 10: The first week is week 1 (already at ₹5). She only makes 9 increases to get to week 10. Do not multiply by 10!
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Solution:
\(t_{10} = a + 9d\)
\(t_{10} = 5 + 9(1.75)\)
\(t_{10} = 5 + 15.75 = 20.75\)
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Select the correct option:
When solving a word problem about "number of rows" or "number of weeks", why must we reject any negative or fractional answers for \(n\)?
Concept:
1
In sequence theory, the domain of the index \(n\) is strictly the set of natural numbers (\(1, 2, 3 \dots\)). This perfectly maps to real-world discrete counting.
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Hint:
1
Think logically: Can you sit in row number -4 in a stadium?
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Mistakes to Avoid:
!
Accepting bad algebra: If you are solving a word problem for "how many rows" and get \(n = -8\), do not write "-8 rows". This is a signal that your algebraic signs were wrong in the previous step!
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Solution:
Because \(n\) counts discrete physical items (like weeks, or rows), it must be a positive, natural integer. Negative or fractional rows do not exist in reality.
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