Consider all natural numbers whose decimal ex | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the set of numbers be $S = \{2, 4, 6, 8, 20, 22, 24, 26, 28, 40, \dots \}$.These numbers are formed using $5$ digits $\{0, 2, 4, 6, 8\}$.Since

Vedclass · Accepted Answer

$\log_5 10$

Vedclass · Answer

(B) Let the set of numbers be $S = \{2, 4, 6, 8, 20, 22, 24, 26, 28, 40, \dots \}$.These numbers are formed using $5$ digits $\{0, 2, 4, 6, 8\}$.Since $0$ cannot be the leading digit,the $n$-th number $a_n$ can be related to the base-$5$ representation.Let $n$ be represented in base $5$ as $(d_k d_{k-1} \dots d_0)_5$. The $n$-th number $a_n$ is obtained by mapping the digits $0, 1, 2, 3, 4$ to $0, 2, 4, 6, 8$ respectively.For large $n$,$a_n \approx 2 \cdot 10^{\log_5 n}$.Taking the logarithm,$\log a_n \approx \log 2 + \log_5 n \cdot \log 10 = \log 2 + \frac{\log n}{\log 5} \cdot \log 10$.Dividing by $\log n$ and taking the limit as $n \rightarrow \infty$:$\lim_{n \rightarrow \infty} \frac{\log a_n}{\log n} = \frac{\log 10}{\log 5} = \log_5 10$.

Consider all natural numbers whose decimal expansion has only the even digits $0, 2, 4, 6, 8$. Suppose these are arranged in increasing order. If $a_n$ denotes the $n$-th number in this sequence,then $\frac{\lim_{n \rightarrow \infty} \log a_n}{\log n}$ equals

Similar Questions

If $\frac{1}{2} \le \log_{0.1} x \le 2$,then:

If ${\log _{10}}3 = 0.477$,then the number of digits in ${3^{40}}$ is

The equation $\log_e x + \log_e(1 + x) = 0$ can be written as

If $\log _{2}\left(9^{x-1}+7\right)-\log _{2}\left(3^{x-1}+1\right)=2$,then $x$ values are

Let $a, b, c$ be real numbers, each greater than $1$, such that $\frac{2}{3} \log _{b} a+\frac{3}{5} \log _{c} b+\frac{5}{2} \log _{a} c=3$. If the value of $b$ is $9$, then the value of $a$ must be

Vedclass Test Series

Exam Paper Generator

Online Exam Module