$$ \frac { 5 ^ { n - 1 } \cdot 2 ^ { n + 2 } } { 10 ^ { n } - 10 ^ { n - 1 } \cdot 2 } $$

Question

UpStudy AI · Accepted Answer

Let's simplify the expression:

$$
\frac{5^{n - 1} \cdot 2^{n + 2}}{10^n - 10^{n - 1} \cdot 2}
$$

### Step 1: Express Everything in Terms of Primes

First, note that $$10 = 2 \times 5$$. We'll rewrite the denominator using this:

$$
10^n = (2 \times 5)^n = 2^n \cdot 5^n
$$
$$
10^{n-1} = (2 \times 5)^{n-1} = 2^{n-1} \cdot 5^{n-1}
$$

So, the denominator becomes:

$$
10^n - 10^{n-1} \cdot 2 = 2^n \cdot 5^n - 2^{n-1} \cdot 5^{n-1} \cdot 2
$$

Simplify the second term in the denominator:

$$
2^{n-1} \cdot 5^{n-1} \cdot 2 = 2^n \cdot 5^{n-1}
$$

Now, rewrite the denominator:

$$
2^n \cdot 5^n - 2^n \cdot 5^{n-1} = 2^n \cdot 5^{n-1} \cdot (5 - 1) = 2^n \cdot 5^{n-1} \cdot 4
$$

So, the denominator simplifies to:

$$
4 \cdot 2^n \cdot 5^{n-1}
$$

### Step 2: Rewrite the Numerator

The numerator is:

$$
5^{n - 1} \cdot 2^{n + 2} = 5^{n-1} \cdot 2^n \cdot 2^2 = 5^{n-1} \cdot 2^n \cdot 4
$$

### Step 3: Form the Simplified Fraction

Now, the expression becomes:

$$
\frac{5^{n-1} \cdot 2^n \cdot 4}{4 \cdot 2^n \cdot 5^{n-1}}
$$

### Step 4: Cancel Common Terms

Notice that $$5^{n-1}$$, $$2^n$$, and $$4$$ appear in both the numerator and the denominator. Cancel them out:

$$
\frac{\cancel{5^{n-1}} \cdot \cancel{2^n} \cdot \cancel{4}}{\cancel{4} \cdot \cancel{2^n} \cdot \cancel{5^{n-1}}} = 1
$$

### Conclusion

After simplifying, the expression equals **1** for all positive integers $$n$$:

$$
\frac{5^{n - 1} \cdot 2^{n + 2}}{10^n - 10^{n - 1} \cdot 2} = 1
$$
The expression simplifies to 1 for all positive integers $$ n $$.

\( \frac { 5 ^ { n - 1 } \cdot 2 ^ { n + 2 } } { 10 ^ { n } - 10 ^ { n - 1 } \cdot 2 } \)