Question

If $R = \frac{30^{65} - 29^{65}}{30^{64} + 29^{64}}$, then:

• A. $0<R \leq 0.1$
• B. $0.1<R \leq 0.5$
• C. $0.5<R \leq 1.0$
• D. $R>1.0$

Hint:

In large power ratios, factor out the largest term to simplify.

Answer 1

The Correct Option is D

To solve the given problem, we need to simplify the expression for \( R \): 

\[ R = \frac{30^{65} - 29^{65}}{30^{64} + 29^{64}} \]

Let's analyze the numerator and the denominator:

• The numerator is \( 30^{65} - 29^{65} \) which can be rewritten using the identity \( a^{n} - b^{n} = (a-b)(a^{n-1} + a^{n-2}b + \ldots + ab^{n-2} + b^{n-1}) \). Here, \( a = 30 \), \( b = 29 \), and \( n = 65 \).
• The denominator is \( 30^{64} + 29^{64} \).

Substituting into the expression for \( R \), we get:

\[ R = \frac{(30 - 29)(30^{64} + 30^{63}\cdot29 + \cdots + 30\cdot29^{63} + 29^{64})}{30^{64} + 29^{64}} \]

Which simplifies to:

\[ R = \frac{1 \times (30^{64} + 30^{63}\cdot29 + \cdots + 30\cdot29^{63} + 29^{64})}{30^{64} + 29^{64}} \]

The top part: \( 30^{64} + 30^{63}\cdot29 + \cdots + 29^{64} \) contains many terms, all of which are positive and greater than zero. Since these terms are more numerous than those in the denominator \((30^{64} + 29^{64})\), we can deduce that:

\[ R > 1 \]

Clearly, the simplified form reveals that \( R \) is greater than 1. Thus the correct answer is:

\[ R > 1.0 \]