JEE Advanced · 2007 · 08 Apr 2007 · Paper 1 · Sequences and Series

More JEE Advanced questions on Sequences and Series

1. Let $\mathbb{R}$ denote the set of all real numbers. Let $f:\mathbb{R}\to \mathbb{R}$ be a function such that $f(x)>0$ for all $x\in \mathbb{R}$, and $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbb{R}$.
   
   Let the real numbers …
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2. Let $\mathrm{75...57}$ denote the $(\mathrm{r}+2)$ digit number where the first and the last digits are 7 and the remaining r digits are 5. Consider the sum $\mathrm{S}=77+757+7557+\dots +75\dots 57$. If $\mathrm{S}=\frac{75\dots 57+\mathrm{m}}{\mathrm{n}}$, where m and n are natur …
   JEE Advanced 2023 Copy linkView solution →
3. Let $l_1,l_2,\dots ,l_{100}$ be consecutive terms of an arithmetic progression with common difference $d_1$, and let $w_1,w_2,\dots ,w_{100}$ be consecutive terms of another arithmetic progression with common difference $d_2$, where $d_1{d}_2=10$. For each …
   JEE Advanced 2022 Copy linkView solution →
4. Let $a_1,a_2,a_3,\dots$ be an arithmetic progression with $a_1=7$ and common difference 8. Let $T_1,T_2,T_3,\dots$ be such that $T_1=3$ and $T_{n+1}-T_n=a_n$ for $n\ge 1$. Then, which of the following is/are TRUE ?
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5. For positive integer n, define
   

   $$f(n)=n+\frac{16+5n-3n^2}{4n+3n^2}+\frac{32+n-3n^2}{8n+3n^2}+\frac{48-3n-3n^2}{12n+3n^2}+\dots +\frac{25n-7n^2}{7n^2}.$$

   
   Then, the value of ${\lim }_{n\to \infty }f(n)$ is equal to
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6. Let
   

   $$\mathrm{M}=\left\{(\mathrm{x},\mathrm{y})\in \mathbb{R}\times \mathbb{R}:{\mathrm{x}}^2+{\mathrm{y}}^2\le {\mathrm{r}}^2\right\},$$

   
   where $\mathrm{r}>0$. Consider the geometric progression …
   JEE Advanced 2021 Copy linkView solution →