List of top Mathematics Questions on Sequence and series asked in JEE Advanced

Let $a_1, a_2, a_3, \ldots$ be an arithmetic progression with $a_1=7$ and common difference 8 Let $T_1, T_2, T_3, \ldots$ be such that $T_1=3$ and $T_{n+1}-T_n=a_n$ for $n \geq 1$ Then, which of the following is/are TRUE ?

Let $\alpha=\displaystyle\sum_{ k =1}^{\infty} \sin ^{2 k }\left(\frac{\pi}{6}\right)$ Let $g:[0,1] \rightarrow R$ be the function defined by $g(x)=2^{\alpha x}+2^{\alpha(1-x)}$. Then, which of the following statements is/are TRUE?

Let $a_1, a_2, a_3, \ldots$ be an arithmetic progression with $a_1=7$ and common difference 8. Let $T_1, T_2, T_3, \ldots$ be such that $T_1=3$ and $T_{n+1}-T_n=a_n$ for $n \geq 1$. Then, which of the following is/are TRUE ?

Let $\alpha=\displaystyle\sum_{ k =1}^{\infty} \sin ^{2 k }\left(\frac{\pi}{6}\right)$ Let $g:[0,1] \rightarrow R$ be the function defined by $g(x)=2^{\alpha x}+2^{\alpha(1-x)}$. Then, which of the following statements is/are TRUE?

For any positive integer $n$, let $S_{n}:(0, \infty) \rightarrow R$ be defined by
$S_{n}(x)=\displaystyle\sum_{k=1}^{n} \cot ^{-1}\left(\frac{1+k(k+1) x^{2}}{x}\right),$
where for any $x \in R, \cot ^{-1}(x) \in(0, \pi)$ and $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then which of the following statements is(are) TRUE?

Consider $M$ with $r=\frac{1025}{513}$. Let $k$ be the number of all those circles $C_{n}$ that are inside $M$. Let $1$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect Then

Consider $M$ with $r=\frac{\left(2^{199}-1\right) \sqrt{2}}{2^{198}}$. The number of all those circles $D_{n}$ that are inside $M$ is

Let
$M=\left\{(x, y) \in R \times R: x^{2}+y^{2} \leq r^{2}\right\}$,
where $r$$>$$0$. Consider the geometric progression $a_{n}=\frac{1}{2^{n-1}}, n=1,2,3, \ldots $ Let $S_{0}=0$ and for $n \geq 1$, let $S_{n}$ denote the sum of the first $n$ terms of this progression. For $n \geq 1$, let $C_{n}$ denote the circle with center $\left(S_{n-1}, 0\right)$ and radius $a_{n}$, and $D_{n}$ denote the circle with center $\left(S_{n-1}, S_{n-1}\right)$ and radius $a_{n}$.