Sequence and Series — NDA Mathematics PYQs

Let $p$, $q$ and $r$ be three unequal numbers such that $p$, $q$ and $r$ are in AP. If $(q-p)$, $(r-q)$ and $p$ are in GP, then $(p+q):(q+r):(r+p)$ equals

If $p$, $g_1$, $g_2$ and $q$ are in GP and $m$ is the arithmetic mean of $p$ and $q$, then $\frac{g_1^2}{g_2}+\frac{g_2^2}{g_1}$ is equal to

The sum of the first $n$ terms of an AP is $3n^2+5n$. If the $m^{\text{th}}$ term of the AP is $68$, then what is the value of $m$ ?

If $p^{\text{th}}$ term of an AP is $k$, then what is the sum of $p^{\text{th}}$ term, $(p+q{)}^{\text{th}}$ term and $(p-q{)}^{\text{th}}$ term of the AP ?

The sum of the first $k$ terms of a series $S$ is $3k^2+5k$. Which one of the following is correct?

The sum of the first $8$ terms of a GP is five times the sum of its first $4$ terms. If $r\ne 1$ is the common ratio, then what is the number of possible real values of $r$?

If $5$th, $7$th and $13$th terms of an AP are in GP, then what is the ratio of its first term to its common difference?

If $p,1,q$ are in AP and $p,2,q$ are in GP, then which of the following statements is/are correct?
I. $p,4,q$ are in HP.
II. $\frac{1}{p},\frac{1}{4},\frac{1}{q}$ are in AP.
Select the answer using the code given below.

If $p^x=q^y=r^z$, where $x$, $y$ and $z$ are in GP, then consider the following statements :
I. $p$, $q$ and $r$ are in AP.
II. $\ln p$, $\ln q$ and $\ln r$ are in GP.
Which of the statements given above is/are correct?

If $(10+{\log }_{10}x)$, $(10+{\log }_{10}y)$ and $(10+{\log }_{10}z)$ are in AP, then consider the following statements :
I. The GM of $x$ and $z$ is $y^2$.
II. The AM of ${\log }_{10}x$ and ${\log }_{10}z$ is ${\log }_{10}y$.
Which of the statements given above is/are correct?