1
WB JEE 2012
+1
-0.33
The value of $$\sum\limits_{r = 2}^\infty {{{1 + 2 + .... + (r - 1)} \over {r!}}}$$
A
e
B
2e
C
$${e \over 2}$$
D
$${{3e} \over 2}$$
2
WB JEE 2024
+1
-0.25

Given an A.P. and a G.P. with positive terms, with the first and second terms of the progressions being equal. If $$a_n$$ and $$b_n$$ be the $$n^{\text {th }}$$ term of A.P. and G.P. respectively then

A
$$a_n>b_n$$ for all $$n>2$$
B
$$a_n< b_n$$ for all $$n>2$$
C
$$a_n=b_n$$ for some $$n>2$$
D
$$a_n=b_n$$ for some odd $$n$$
3
WB JEE 2024
+1
-0.25

If for the series $$a_1, a_2, a_3$$, ...... etc, $$\mathrm{a}_{\mathrm{r}}-\mathrm{a}_{\mathrm{r}+\mathrm{i}}$$ bears a constant ratio with $$\mathrm{a}_{\mathrm{r}} \cdot \mathrm{a}_{\mathrm{r}+1}$$; then $$\mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3 \ldots .$$. are in

A
A.P.
B
G.P.
C
H.P.
D
Any other series
4
WB JEE 2024
+2
-0.5

If $$\alpha_1, \alpha_2, \ldots, \alpha_n$$ are in A.P. with common difference $$\theta$$, then the sum of the series $$\sec \alpha_1 \sec \alpha_2+\sec \alpha_2 \sec \alpha_3+\ldots .+\sec \alpha_{n-1} \sec \alpha_n=k\left(\tan \alpha_n-\tan \alpha_1\right)$$ where $$\mathrm{k}=$$

A
$$\sin \theta$$
B
$$\cos \theta$$
C
$$\sec \theta$$
D
$$\operatorname{cosec} \theta$$
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