If $R = \frac{30^{65} - 29^{65}}{30^{64} + 29^{64}}$, then:
Show Hint
- $0<R \leq 0.1$
- $0.1<R \leq 0.5$
- $0.5<R \leq 1.0$
- $R>1.0$
The Correct Option is D
Solution and Explanation
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:
\[ R > 1.0 \]
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