Sequences and Series

If $x, y, z$ are in GP, then which of the following is/are correct? $\ln (3 x), \ln (3 y), \ln (3 z)$ are in AP $x y z+\ln (), x y z+\ln (y), x y z+\ln (z)$ are in HP Select the correct answer using the code given below.

If $a_n=n(n!)$, then what is $a_1+a_2+a_3+\ldots+a_{10}$ equal to?

If $2^{\frac{1}{c}}, 2^{\frac{b}{a c}}, 2^{\frac{1}{a}}$ are in GP, then which one of the following is correct?

Let $m=77^n$. The index $n$ is given a positive integral value at random. What is the probability that the value of $m$ will have 1 in the units place?

Paragraph: Given that $m(\theta)=\cot ^2 \theta+n^2 \tan ^2 \theta+2 n$, where $n$ is a fixed positive real number. Question: What is the least value of $\mathrm{m}(\theta)$ ?

For the following two (02) items: Let $(6 + 10 + 14 + \ldots\text{ up to }m\text{ terms}) = (1 + 3 + 5 + 7 + \ldots\text{ up to }n\text{ terms})$, where $m < 25$ and $n < 25$. What is the relation between $m$ and $n$?

If $p=(1111 \ldots$ up to $n$ digits $)$, then what is the value of $9 p^2+p$ ?

If $g$ is the geometric mean of $2,4,8,16,32,64,128,256,512$, 1024, then which one of the following is correct?

What is the mean of natural numbers contained in the interval $[15,64]$ ?

If the 5th term of an AP is $\frac{1}{10}$ and its 10 th term is $\frac{1}{5}$, then what is the sum of first 50 terms?

If $a, b, c$ are in GP where $a \gt 0, b \gt 0, c \gt 0$, then which of the following are correct? $a^2, b^2, c^2$ are in GP $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in GP $\sqrt{a}, \sqrt{b}, \sqrt{c}$ are in GP…

If G is the geometric mean of numbers $1,2,2^2, 2^3, \ldots, 2 n^{-1}$, then what is the value of $1+2 \log _2 \mathrm{G}$ ?