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

Question

UpStudy AI · Accepted Answer

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:**

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

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