JEE Main · 2023 · 13 Apr 2023 · Shift 1 · Sequences and Series

More JEE Main questions on Sequences and Series

1. Let ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots$. be a G.P. of increasing positive terms such that $a_2\cdot a_3\cdot a_4=64$ and $a_1+a_3+a_5=\frac{813}{7}$.
   
   Then ${\mathrm{a}}_3+{\mathrm{a}}_5+{\mathrm{a}}_7$ is equal to :
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2. Let ${\mathrm{a}}_1=1$ and for $\mathrm{n}\ge 1,{\mathrm{a}}_{\mathrm{n}+1}=\frac{1}{2}{a}_n+\frac{n^2-2n-1}{n^2(n+1{)}^2}$. Then $\left|{\sum }_{n=1}^{\infty }\left(a_n-\frac{2}{n^2}\right)\right|$ is equal to
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3. The positive integer n, for which the solutions of the equation
   

   $$x(x+2)+(x+2)(x+4)+\dots +(x+2n-2)(x+2n)=\frac{8n}{3}$$

   
   are two consecutive even integers, is :-
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4. Let $a_1,\frac{a_2}{2},\frac{a_3}{2^2},\dots ,\frac{a_{10}}{2^9}$ be a G.P. of common ratio $\frac{1}{\sqrt{2}}$. If $a_1+a_2+\dots +a_{10}=62$, then $a_1$ is equal to :
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5. If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4, then the sum of its first twelve terms is
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6. Suppose $\mathrm{a},\mathrm{b},\mathrm{c}$ are in A.P. and ${\mathrm{a}}^2,2{\mathrm{b}}^2,{\mathrm{c}}^2$ are in G.P. If $\mathrm{a}<\mathrm{b}<\mathrm{c}$ and $\mathrm{a}+\mathrm{b}+\mathrm{c}=1$, then $9\left({\mathrm{a}}^2+{\mathrm{b}}^2+c^2\right)$ is equa …
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