Sequence and Series — Class 12 Mathematics PYQs

If $a,b,c$ are in AP; $b,c,d$ are in GP; $c,d,e$ are in HP, then which of the following is/are correct?

1. $a,c$ and $e$ are in GP
2. $\frac{1}{a},\frac{1}{c},\frac{1}{e}$ are in GP

Select the correct answer using the code given below:

Direction : Consider the following for the two (02) items that follow :

The product of 5 consecutive terms of an AP is 229635. The first, second and fifth terms are in GP.

What is the sum of all five terms?

Direction : Consider the following for the two (02) items that follow :

The roots of the quadratic equation
$$a^2(b^2-c^2)x^2+b^2(c^2-a^2)x+c^2(a^2-b^2)=0$$
are equal ($a^2\ne b^2\ne c^2$).

Which one of the following statements is correct?

If $1!+3!+5!+7!+\dots +199!$ is divided by $24$, what is the remainder ?

If the $n^{\text{th}}$ term of a sequence is $\frac{2n+5}{7}$, then what is the sum of its first $140$ terms ?

If $(a+b)$, $2b$, $(b+c)$ are in HP, then which one of the following is correct ?

Let $t_1,t_2,t_3\dots$ be in GP. What is $(t_1{t}_3\dots t_{21}{)}^{\frac{1}{11}}$ equal to ?

If ${\log }_{x}a$, $a^x$ and ${\log }_{b}x$ are in GP, then what is $x$ equal to?

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

The first and the second terms of an AP are $\frac{5}{2}$ and $\frac{23}{12}$ respectively. If $n^{\text{th}}$ term is the largest negative term, what is the value of $n$?