As we reach the peak of the End of Chapter exercises, the problems stop asking for simple term calculations and start demanding a deep synthesis of both Arithmetic and Geometric logic.
You will need to use symmetric variable assumptions for GPs (just like we did for APs) to easily cancel out ratios, and you will even encounter sequences that seamlessly transform from APs into GPs. Let's tackle these logic puzzles!
You will need to use symmetric variable assumptions for GPs (just like we did for APs) to easily cancel out ratios, and you will even encounter sequences that seamlessly transform from APs into GPs. Let's tackle these logic puzzles!
1 Symmetric GP Assumptions
2 The AP/GP Transformation
★ Advanced Ratio Proof
Select the correct option:
When solving a Geometric Progression problem that provides the product of three unknown terms, which algebraic assumption is the most mathematically efficient?
Concept:
1
Just like we use symmetry in APs to cancel out addition, we use symmetry in GPs to cancel out multiplication.
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Hint:
1
Which set of terms will eliminate \(r\) completely when they are multiplied together?
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Mistakes to Avoid:
!
Relying on the explicit formula: You can use \(a, ar, ar^2\), but you will create the equation \(a^3 r^3 = X\), which is a nightmare to separate. Symmetric terms are a critical shortcut!
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Solution:
The optimal assumption is \(\frac{a}{r}, a, ar\).
Product = \(\frac{a}{r} \cdot a \cdot ar = a^3\).
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Select the correct option:
Using the symmetric assumption \(\frac{a}{r}, a, ar\) for a Geometric Progression, what is the exact product of these three terms?
Concept:
1
Fractions multiply straight across. A variable in the numerator and the same variable in the denominator cancel to 1.
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Hint:
1
\(\frac{a}{r} \times \frac{a}{1} \times \frac{ar}{1}\). Combine the tops and bottoms!
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Mistakes to Avoid:
!
Adding instead of multiplying: \(\frac{a}{r} + a + ar\) cannot be simplified to \(a^3\). This trick ONLY works for the product!
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Solution:
\(\frac{a}{r} \cdot a \cdot ar = \frac{a \cdot a \cdot a \cdot r}{r} = a^3\).
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Select the correct option:
If the Arithmetic Mean of two positive numbers is exactly equal to their Geometric Mean, what must be true about the two numbers?
Concept:
1
Arithmetic Mean = \(\frac{a+b}{2}\). Geometric Mean = \(\sqrt{ab}\).
2
Set them equal: \(\frac{a+b}{2} = \sqrt{ab}\) and solve algebraically.
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Hint:
1
Multiply by 2: \(a+b = 2\sqrt{ab}\).
2
Square both sides: \((a+b)^2 = 4ab \implies a^2 + 2ab + b^2 = 4ab\).
3
\(a^2 - 2ab + b^2 = 0 \implies (a-b)^2 = 0\).
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Mistakes to Avoid:
!
Assuming AM is always larger: The AM is strictly greater than the GM (\(AM > GM\)) unless the numbers are perfectly identical.
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Solution:
If \(AM = GM\):
\(\frac{a+b}{2} = \sqrt{ab}\)
\(a+b - 2\sqrt{ab} = 0\)
\((\sqrt{a} - \sqrt{b})^2 = 0 \implies \sqrt{a} = \sqrt{b} \implies a = b\).
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Select the correct option:
What is the Arithmetic Mean (AM) and the Geometric Mean (GM) of the numbers 4 and 16?
Concept:
1
Arithmetic Mean = \(\frac{x+y}{2}\).
2
Geometric Mean = \(\sqrt{xy}\).
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Hint:
1
Calculate \(\frac{4+16}{2}\).
2
Calculate \(\sqrt{4 \times 16} = \sqrt{64}\).
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Mistakes to Avoid:
!
Forgetting the square root: A common error is multiplying the numbers but forgetting to root them. \(64\) is the product, \(8\) is the mean!
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Solution:
\(AM = \frac{4+16}{2} = 10\)
\(GM = \sqrt{4 \times 16} = \sqrt{64} = 8\)
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Select the correct option:
If a problem states that \(d = 2a\) in an Arithmetic Progression, what does the algebraic fraction \(\frac{t_m}{t_n}\) simplify to?
Concept:
1
Write the term ratio: \(\frac{a + (m-1)d}{a + (n-1)d}\).
2
Substitute \(2a\) wherever you see \(d\), then factor out \(a\) to cancel it.
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Hint:
1
Numerator: \(a + (m-1)(2a) = a + 2am - 2a = 2am - a\).
2
Factor out \(a\): \(a(2m - 1)\).
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Mistakes to Avoid:
!
Confusing sums and terms: \(m^2:n^2\) is the ratio of the SUMS (\(S_m/S_n\)) under this condition, but the question specifically asks for the ratio of the TERMS (\(t_m/t_n\)).
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Solution:
\(\frac{t_m}{t_n} = \frac{a + (m-1)2a}{a + (n-1)2a}\)
\(= \frac{a(1 + 2m - 2)}{a(1 + 2n - 2)}\)
\(= \frac{2m - 1}{2n - 1}\)
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