1
JEE Main 2022 (Online) 30th June Morning Shift
+4
-1

The value of $$1 + {1 \over {1 + 2}} + {1 \over {1 + 2 + 3}} + \,\,....\,\, + \,\,{1 \over {1 + 2 + 3 + \,\,.....\,\, + \,\,11}}$$ is equal to:

A
$${{20} \over {11}}$$
B
$${{11} \over {6}}$$
C
$${{241} \over {132}}$$
D
$${{21} \over {11}}$$
2
JEE Main 2022 (Online) 29th June Evening Shift
+4
-1

The sum of the infinite series $$1 + {5 \over 6} + {{12} \over {{6^2}}} + {{22} \over {{6^3}}} + {{35} \over {{6^4}}} + {{51} \over {{6^5}}} + {{70} \over {{6^6}}} + \,\,.....$$ is equal to :

A
$${{425} \over {216}}$$
B
$${{429} \over {216}}$$
C
$${{288} \over {125}}$$
D
$${{280} \over {125}}$$
3
JEE Main 2022 (Online) 29th June Morning Shift
+4
-1

Let $$\{ {a_n}\} _{n = 0}^\infty$$ be a sequence such that $${a_0} = {a_1} = 0$$ and $${a_{n + 2}} = 2{a_{n + 1}} - {a_n} + 1$$ for all n $$\ge$$ 0. Then, $$\sum\limits_{n = 2}^\infty {{{{a_n}} \over {{7^n}}}}$$ is equal to:

A
$${6 \over {343}}$$
B
$${7 \over {216}}$$
C
$${8 \over {343}}$$
D
$${{49} \over {216}}$$
4
JEE Main 2022 (Online) 28th June Evening Shift
+4
-1

If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is :

A
21
B
22
C
23
D
24
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