Sequence and Series — NDA Mathematics PYQs

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?

How many terms of the series $1+3+5+7+\dots$ amount to a sum equal to $12345678987654321$?

How many terms are identical in the two APs $19,21,23,\dots$ up to $110$ terms and $19,22,25,28,\dots$ up to $75$ terms?

Consider the following for the two (02) items that follow :
Let $p=\sum _{j=1}^n{\log }_{10}{2}^j$ and $q=\sum _{j=1}^n{\log }_{10}{5}^j$.

If $p+q=66$, then which one of the following is correct?

Consider the following for the two (02) items that follow :
Let $(6+10+14+\dots$ up to $m$ terms$)=(1+3+5+7+\dots$ up to $n$ terms$)$ where $m<25$ and $n<25$.

What is the relation between $m$ and $n$?

Consider the following for the two (02) items that follow :
Let $(6+10+14+\dots$ up to $m$ terms$)=(1+3+5+7+\dots$ up to $n$ terms$)$ where $m<25$ and $n<25$.

How many values of $m$ are possible?