1
JEE Main 2023 (Online) 31st January Morning Shift
Numerical
+4
-1

Let $$a_{1}, a_{2}, \ldots, a_{n}$$ be in A.P. If $$a_{5}=2 a_{7}$$ and $$a_{11}=18$$, then

$$12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}}+\frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}}+\ldots+\frac{1}{\sqrt{a_{17}}+\sqrt{a_{18}}}\right)$$ is equal to ____________.

2
JEE Main 2023 (Online) 30th January Evening Shift
Numerical
+4
-1
Out of Syllabus
The $8^{\text {th }}$ common term of the series

\begin{aligned} & S_1=3+7+11+15+19+\ldots . . \\\\ & S_2=1+6+11+16+21+\ldots . . \end{aligned}

is :
3
JEE Main 2023 (Online) 30th January Morning Shift
Numerical
+4
-1
Out of Syllabus

Let $$\sum_\limits{n=0}^{\infty} \frac{\mathrm{n}^{3}((2 \mathrm{n}) !)+(2 \mathrm{n}-1)(\mathrm{n} !)}{(\mathrm{n} !)((2 \mathrm{n}) !)}=\mathrm{ae}+\frac{\mathrm{b}}{\mathrm{e}}+\mathrm{c}$$, where $$\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathbb{Z}$$ and $$e=\sum_\limits{\mathrm{n}=0}^{\infty} \frac{1}{\mathrm{n} !}$$ Then $$\mathrm{a}^{2}-\mathrm{b}+\mathrm{c}$$ is equal to ____________.

4
JEE Main 2023 (Online) 29th January Evening Shift
Numerical
+4
-1
Out of Syllabus

Let $$a_1=b_1=1$$ and $${a_n} = {a_{n - 1}} + (n - 1),{b_n} = {b_{n - 1}} + {a_{n - 1}},\forall n \ge 2$$. If $$S = \sum\limits_{n = 1}^{10} {{{{b_n}} \over {{2^n}}}}$$ and $$T = \sum\limits_{n = 1}^8 {{n \over {{2^{n - 1}}}}}$$, then $${2^7}(2S - T)$$ is equal to ____________.

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