Sequence-series – JEE Advanced Previous Year Questions with Solutions
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Q. If the sum of first $n$ terms of an A.P. is $\mathrm{c} n^{2},$ then the sum of squares of these $n$ terms is(A) $\frac{\mathrm{n}\left(4 \mathrm{n}^{2}-1\right) \mathrm{c}^{2}}{6}$(B) $\frac{\mathrm{n}\left(4 \mathrm{n}^{2}+1\right) \mathrm{c}^{2}}{3}$(C) $\frac{\mathrm{n}\left(4 \mathrm{n}^{2}-1\right) \mathrm{c}^{2}}{3}$(D) $\frac{n\left(4 n^{2}+1\right) c^{2}}{6}$ [JEE 2009, 3 (–1)]

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Sol. (C)$\mathrm{C}$$\mathrm{S}_{\mathrm{n}}=\mathrm{cn}^{2}$$\mathrm{S}_{\mathrm{n}-1}=\mathrm{c}(\mathrm{n}-1)^{2}$$\mathrm{T}_{\mathrm{n}}=\mathrm{S}_{\mathrm{n}}-\mathrm{S}_{\mathrm{n}-1}=\mathrm{c}(2 \mathrm{n}-1)$$\mathrm{T}_{\mathrm{n}}^{\prime}=\mathrm{T}_{\mathrm{n}}^{2}=\mathrm{c}^{2}\left(4 \mathrm{n}^{2}-4 \mathrm{n}+1\right)$$\sum \mathrm{T}_{\mathrm{n}}^{\prime}=\mathrm{nc}^{2}\left(\frac{4 \mathrm{n}^{2}-1}{3}\right) Q. Let a_{1}, a_{2}, a_{3}, \ldots \ldots \ldots a_{11} be real numbers satisfying\mathrm{a}_{1}=15,27-2 \mathrm{a}_{2}>0 and \mathrm{a}_{\mathrm{k}}=2 \mathrm{a}_{\mathrm{k}-1}-\mathrm{a}_{\mathrm{k}-2} for \mathrm{k}=3,4 \ldots \ldots \ldots 11If \frac{a_{1}^{2}+a_{2}^{2}+\ldots \ldots+a_{11}^{2}}{11}=90, then the value of \frac{a_{1}+a_{2}+\ldots \ldots+a_{11}}{11} is equal to [JEE 2010,3+3] Download eSaral App for Video Lectures, Complete Revision, Study Material and much more... Sol. 0 Q. The minimum value of the sum of real numbers a^{-5}, a^{-4}, 3 a^{-3}, 1, a^{8} and a^{10} with a>0 is [JEE 2011, 4] Download eSaral App for Video Lectures, Complete Revision, Study Material and much more... Sol. 8As a>0and all the given terms are positivehence considering A.M. \geq G.M. for given numbers :\frac{\mathrm{a}^{-5}+\mathrm{a}^{-4}+\mathrm{a}^{-3}+\mathrm{a}^{-3}+\mathrm{a}^{-3}+\mathrm{a}^{8}+\mathrm{a}^{10}}{7} \geq\left(\mathrm{a}^{-5} \cdot \mathrm{a}^{-4} \cdot \mathrm{a}^{-3} \cdot \mathrm{a}^{-3} \cdot \mathrm{a}^{-3} \cdot \mathrm{a}^{10}\right)^{\frac{1}{7}}$$\Rightarrow \frac{\mathrm{a}^{-5}+\mathrm{a}^{-4}+\mathrm{a}^{-3}+\mathrm{a}^{-3}+\mathrm{a}^{-3}+\mathrm{a}^{8}+\mathrm{a}^{10}}{7} \geq 1 \quad \Rightarrow\left(\mathrm{a}^{-5}+\mathrm{a}^{-4}+3 \mathrm{a}^{-3}+\mathrm{a}^{-3}+\mathrm{a}^{10}\right)_{\min }=7$where $\mathrm{a}^{-5}=\mathrm{a}^{-4}=\mathrm{a}^{-3}=\mathrm{a}^{8}=\mathrm{a}^{10}$ i.e. $\mathrm{a}=1$$\Rightarrow\left(\mathrm{a}^{-5}+\mathrm{a}^{-4}+3 \mathrm{a}^{-3}+\mathrm{a}^{8}+\mathrm{a}^{10}+1\right)_{\min }=8 \quad when \mathrm{a}=1 Q. Let a_{1}, a_{2}, a_{3}, \ldots \ldots \ldots, a_{100} be an arithmetic progression with a_{1}=3 and S_{p}=\sum_{i=1}^{p} a_{i}, 1 \leq p \leq 100 . For any integer n with 1 \leq n \leq 20, let m=5 n . If \frac{S_{m}}{S_{n}} does not depend on n, then a_{2} is [JEE 2011, 4] Download eSaral App for Video Lectures, Complete Revision, Study Material and much more... Sol. 9 or 39 or 3 (Comment : The information about the common difference i.e. zero or non-zero is not given in the question. Hence there are two possible answers)Consider \mathrm{d} \neq 0 the solution is\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots \ldots \ldots \ldots ., \mathrm{a}_{100} \rightarrow \mathrm{AP}$$\mathrm{a}_{1}=3 ; \quad \mathrm{S}_{\mathrm{p}}=\sum_{\mathrm{i}=1}^{\mathrm{p}} \mathrm{a}_{\mathrm{i}} \quad 1 \leq \mathrm{n} \leq 20$$\mathrm{m}=5 \mathrm{n}$$\frac{\mathrm{S}_{\mathrm{m}}}{\mathrm{S}_{\mathrm{n}}}=\frac{\frac{\mathrm{m}}{2}\left[2 \mathrm{a}_{1}+(\mathrm{m}-1) \mathrm{d}\right]}{\frac{\mathrm{n}}{2}\left[2 \mathrm{a}_{1}+(\mathrm{n}-1) \mathrm{d}\right]}$$\frac{\mathrm{S}_{\mathrm{m}}}{\mathrm{S}_{\mathrm{n}}}=\frac{5\left[\left(2 \mathrm{a}_{1}-\mathrm{d}\right)+5 \mathrm{nd}\right]}{\left[\left(2 \mathrm{a}_{1}-\mathrm{d}\right)+\mathrm{nd}\right]} for \frac{\mathrm{S}_{\mathrm{m}}}{\mathrm{S}_{\mathrm{n}}} to be independent of \mathrm{n}$$\begin{array}{ll}{\therefore 2 \mathrm{a}_{1}-\mathrm{d}=0} & {\Rightarrow \mathrm{d}=2 \mathrm{a}_{1} \Rightarrow \mathrm{d}=6 \Rightarrow \mathrm{a}_{2}=9} \\ {\text { If } \mathrm{d}=0 \quad \Rightarrow} & {\mathrm{a}_{2}=\mathrm{a}_{1}=3}\end{array}$

Q. Let $a_{1}, a_{2}, a_{3}, \ldots .$ be in harmonic progression with $a_{1}=5$ and $a_{20}=25 .$ The least positive integer n for which $a_{n}<0$ is(A) 22          (B) 23          (C) 24             (D) 25 [JEE 2012, 3 (–1)]

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Sol. (D)$a_{1}, a_{2}, a_{3} \dots \ldots . .$ be in $\mathrm{H.P} \Rightarrow \frac{1}{\mathrm{a}_{1}}, \frac{1}{\mathrm{a}_{2}}, \frac{1}{\mathrm{a}_{3}} \ldots$ be in A.P.in A.P. $\mathrm{T}_{1}=\frac{1}{\mathrm{a}_{1}}=\frac{1}{5}$ and $\mathrm{T}_{20}=\frac{1}{\mathrm{a}_{20}}=\frac{1}{25}$$\therefore \mathrm{T}_{20}=\mathrm{T}_{1}+19 \mathrm{d}$$\frac{1}{25}=\frac{1}{5}+19 \mathrm{d} \Rightarrow \mathrm{d}=-\frac{4}{19 \times 25}$$\mathrm{T}_{\mathrm{n}}=\mathrm{T}_{1}+(\mathrm{n}-1) \mathrm{d}<0 \Rightarrow \frac{1}{5}-\frac{(\mathrm{n}-1) \cdot 4}{19 \times 25}<0 \quad \Rightarrow \frac{1}{5}<\frac{4(\mathrm{n}-1)}{25 \times 19} \Rightarrow \frac{5 \times 19}{4}+1<\mathrm{n} \Rightarrow \frac{99}{4}<\mathrm{n}$$\Rightarrow$ least positive integer $n$ is 25

Q. Let $\mathrm{S}_{\mathrm{n}}=\sum_{\mathrm{k}=1}^{4 \mathrm{n}}(-1)^{\frac{\mathrm{k}(\mathrm{k}+1)}{2}} \mathrm{k}^{2} .$ Then $\mathrm{S}_{\mathrm{n}}$ can take value(s)(A) 1056        (B) 1088            (C) 1120         (D) 1332 [JEE-Advanced 2013, 4, (–1)]

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Sol. (A,D)$\mathrm{S}_{\mathrm{n}}=-1^{2}-2^{2}+3^{2}+4^{2}-5^{2}-6^{2}+7^{2}+8^{2} \ldots \ldots \ldots$$\mathrm{S}_{\mathrm{n}}^{\mathrm{n}}=\left(3^{2}-1^{2}\right)+\left(4^{2}-2^{2}\right)+\ldots \ldots \ldots \ldots$$\mathrm{S}_{\mathrm{n}}=2(1+2+3+\ldots \ldots+4 \mathrm{n})=\frac{2(4 n)(4 \mathrm{n}+1)}{2}$$\mathrm{S}_{\mathrm{n}}=4 \mathrm{n}(4 \mathrm{n}+1)$$\mathrm{S}_{\mathrm{n}}=4 \mathrm{n}(4 \mathrm{n}+1)=1056$ is possible when $\mathrm{n}=8$$4 n(4 n+1)=1088$ not possible$4 n(4 n+1)=1120$ not possible$4 n(4 n+1)=1332$ possible when $n=9$

Q. A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller to the numbers on the removed cards is k, then k – 20 = [JEE-Advanced 2013, 4, (–1)]

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Sol. 5When 1 and 2 are removed from numbers 1 to n then we get maximum possible sum of remaining numbers and when n –1,n are removed then get minimum possible sum of remaining numbers.

Q. Let a, b, c be positive integers such that $\frac{b}{a}$ is an integer. If $a, b, c$ are in geometric progression and the arithmetic mean of $a, b, c$ is $b+2,$ then the value of $\frac{a^{2}+a-14}{a+1}$ is [JEE(Advanced)-2014, 3]

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Sol. 4

Q. Let $b_{i}>1$ for $i=1,2, \ldots \ldots, 101 .$ Suppose $\log _{e} b_{1}, \log _{e} b_{2}, \ldots \ldots, \log b_{10}$ are in Arithmetic Progression (A.P.) with the common difference log $2 .$ Suppose $a_{1}, a_{2}, \ldots . ., a_{101}$ are in A.P. such that $a_{1}=$ $b_{1}$ and $a_{51}=b_{51}$. If $t=b_{1}+b_{2}+\ldots . .+b_{51}$ and $s=a_{1}+a_{2}+\ldots . .+a_{51}$ then(A) $\mathrm{s}>\mathrm{t}$ and $\mathrm{a}_{101}>\mathrm{b}_{101}$(B) $\mathrm{s}>\mathrm{t}$ and $\mathrm{a}_{101}<\mathrm{b}_{101}$(C) $\mathrm{s}<\mathrm{t}$ and $\mathrm{a}_{101}>\mathrm{b}_{101}$(D) $\mathrm{s}<\mathrm{t}$ and $\mathrm{a}_{101}<\mathrm{b}_{101}$ [JEE(Advanced)-2016]

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Sol. B

Q. The sides of the right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side ? [JEE(Advanced)-2017]

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Sol. 6

Q. Let $X$ be the set consisting of the first 2018 terms of the arithmetic progression $1,6,11, \ldots .$ , and $Y$ be the set consisting of the first 2018 terms of the arithmetic progression $9,16,23,$ $\ldots \ldots$ Then, the number of elements in the set $X \cup Y$ is [JEE(Advanced)-2018]

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Sol. 3748

• May 6, 2021 at 5:58 pm

good

• May 6, 2021 at 5:57 pm

out

• May 6, 2021 at 5:58 pm

out

• January 8, 2022 at 2:09 pm

• November 17, 2020 at 11:02 pm

Very good explanation

• October 22, 2020 at 9:29 pm

Good question..
Hots questions

• October 22, 2020 at 12:14 pm

E saral waale bhaiyya ya sir mene aapko yt pe dekha hai aur in questions me kuch questions nahi dikhre saaf se..
Wo do lines ek ke upr ek aari hai unme to plzz is ko theek krdo aap

• October 7, 2020 at 3:11 pm

• September 16, 2020 at 11:21 pm

gracias

• July 28, 2020 at 6:38 pm

All the PYQs are not there…Your content is seriously very limited

• July 20, 2020 at 8:46 pm

Nice

• July 1, 2020 at 5:35 pm

2nd qūêstitíòñ is hard to read

• June 28, 2020 at 8:15 am