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(a) \( \frac{2.3^{n+1}+3^{n+2}}{2 \cdot 3^{n+5}+3^{n}} \)

Ask by Salazar Tucker. in South Africa
Jan 19,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

The simplified form of the expression is \( \frac{15}{487} \).

Solution

To simplify the expression: \[ \frac{2 \cdot 3^{n+1} + 3^{n+2}}{2 \cdot 3^{n+5} + 3^{n}} \] **Step 1: Factor Common Terms in the Numerator** Factor out \(3^{n+1}\) from the numerator: \[ 2 \cdot 3^{n+1} + 3^{n+2} = 3^{n+1} (2 + 3) = 3^{n+1} \cdot 5 \] **Step 2: Factor Common Terms in the Denominator** Factor out \(3^{n}\) from the denominator: \[ 2 \cdot 3^{n+5} + 3^{n} = 3^{n} (2 \cdot 3^{5} + 1) = 3^{n} (2 \cdot 243 + 1) = 3^{n} \cdot 487 \] **Step 3: Simplify the Fraction** Now, the expression becomes: \[ \frac{3^{n+1} \cdot 5}{3^{n} \cdot 487} \] Simplify by canceling \(3^{n}\): \[ \frac{3^{n} \cdot 3 \cdot 5}{3^{n} \cdot 487} = \frac{3 \cdot 5}{487} = \frac{15}{487} \] **Final Answer:** \[ \frac{15}{487} \]

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The Deep Dive

To simplify the given expression \( \frac{2.3^{n+1}+3^{n+2}}{2 \cdot 3^{n+5}+3^{n}} \), let's manipulate the numerator and denominator. Starting with the numerator: \[ 2.3^{n+1} + 3^{n+2} = 2 \cdot 3^{n+1} + 3^{n+2} = 3^{n+1} (2 + 3) = 3^{n+1} \cdot 5 \] Now for the denominator: \[ 2 \cdot 3^{n+5} + 3^{n} = 3^{n} (2 \cdot 3^{5} + 1) = 3^{n} (486) \quad \text{(since \(3^5 = 243\))} \] The expression now looks like this: \[ \frac{3^{n+1} \cdot 5}{3^{n} \cdot 486} \] Now we can simplify: \[ = \frac{5 \cdot 3^{n+1}}{486 \cdot 3^{n}} = \frac{5 \cdot 3}{486} = \frac{15}{486} \] This can be further simplified: \[ = \frac{5}{162} \] Thus, the simplified expression is: \[ \frac{5}{162} \]

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