SEQUENCES AND SERIES - Pyqs

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Q1
If the sum of the first \(n\) terms of an A.P. is \(3n^2+5n\), then its \(n\)th term is
(Exam: IIT-JEE Year: 1998)
(A) \(6n+5\)
(B) \(6n+2\)
(C) \(6n-1\)
(D) \(3n+5\)
Q2
The number of terms in the A.P. \(5,9,13,\dots,405\) is
(Exam: AIEEE Year: 2006)
(A) 50
(B) 51
(C) 100
(D) 101
Q3
If the A.M. and G.M. of two positive numbers are \(10\) and \(8\) respectively, the numbers are
(Exam: NEET Year: 2014)
(A) \(6,14\)
(B) \(8,12\)
(C) \(4,16\)
(D) \(2,18\)
Q4
The sum of an infinite G.P. with first term \(3\) and ratio \(\frac12\) is
(Exam: IIT-JEE Year: 1995)
(A) 3
(B) 6
(C) 9
(D) 12
Q5
If \(a,b,c\) are in G.P. and \(a+b+c=21\), then \(b\) equals
(Exam: KVPY Year: 2012)
(A) 6
(B) 7
(C) 8
(D) 9
Q6
The sum of first \(n\) odd natural numbers is
(Exam: Olympiad Year: 2001)
(A) \(n(n-1)\)
(B) \(n^2\)
(C) \(2n^2\)
(D) \(n(n+1)\)
Q7
If the 5th term of an A.P. is 20 and 9th term is 36, the first term is
(Exam: IIT-JEE Year: 2003)
(A) 4
(B) 5
(C) 6
(D) 8
Q8
The value of \(1+\frac12+\frac14+\frac18+\cdots\) is
(Exam: BITSAT Year: 2010)
(A) 1
(B) 2
(C) 3
(D) 4
Q9
If \(a_n=\frac{1}{n(n+1)}\), then \(\sum_{k=1}^n a_k\) equals
(Exam: IIT-JEE Year: 2001)
(A) \(\frac{n}{n+1}\)
(B) \(\frac{1}{n+1}\)
(C) \(1-\frac{1}{n}\)
(D) \(\frac{n+1}{n}\)
Q10
If the common ratio of a G.P. is negative, then
(Exam: NEET Year: 2016)
(A) all terms are positive
(B) terms alternate in sign
(C) all terms are negative
(D) sum is always finite
Q11
The arithmetic mean between \(4\) and \(20\) is
(Exam: State Engg. Year: 2008)
(A) 10
(B) 11
(C) 12
(D) 14
Q12
The geometric mean between \(4\) and \(16\) is
(Exam: NEET Year: 2011)
(A) 6
(B) 7
(C) 8
(D) 10
Q13
If \(S_n=2^n-1\), then the sequence is
(Exam: IIT-JEE Year: 1997)
(A) A.P.
(B) G.P.
(C) H.P.
(D) neither
Q14
The sum of first \(n\) terms of A.P. with \(a=2,d=3\) is
(Exam: AIEEE Year: 2005)
(A) \(n(3n+1)\)
(B) \(\frac{n}{2}(4+3n)\)
(C) \(n(2+3n)\)
(D) \(\frac{n}{2}(4+3(n-1))\)
Q15
If the sum of infinite G.P. is 4 and first term is 2, the common ratio is
(Exam: IIT-JEE Year: 1994)
(A) \(\frac12\)
(B) \(\frac14\)
(C) \(-\frac12\)
(D) \(-\frac14\)
Q16
The 10th term of the sequence \(n^2\) is
(Exam: Olympiad Year: 2004)
(A) 90
(B) 95
(C) 100
(D) 110
Q17
If \(a,b,c\) are in A.P. and \(b,c,d\) in G.P., then \(a,c,d\) are
(Exam: IIT-JEE Year: 2007)
(A) in A.P.
(B) in G.P.
(C) equal
(D) none
Q18
The sum \(1^2+2^2+\cdots+n^2\) equals
(Exam: NEET Year: 2013)
(A) \(\frac{n(n+1)}{2}\)
(B) \(\frac{n(n+1)(2n+1)}{6}\)
(C) \(\frac{n^2(n+1)}{2}\)
(D) \(\frac{n(n+1)(n+2)}{6}\)
Q19
If the middle term of an odd-term G.P. is \(m\), then the product of all terms is
(Exam: IIT-JEE Year: 1999)
(A) \(m\)
(B) \(m^n\)
(C) \(m^{\text{number of terms}}\)
(D) \(m^2\)
Q20
The harmonic mean of \(a\) and \(b\) is
(Exam: State Engg. Year: 2009)
(A) \(\frac{a+b}{2}\)
(B) \(\sqrt{ab}\)
(C) \(\frac{2ab}{a+b}\)
(D) \(\frac{ab}{2}\)
Q21
If the sum of the first \(n\) terms of a G.P. is \(S_n=3(2^n-1)\), then the first term is
(Exam: IIT-JEE Year: 1996)
(A) 1
(B) 2
(C) 3
(D) 6
Q22
The number of arithmetic means between 7 and 77 such that the sequence is an A.P. with common difference 7 is
(Exam: AIEEE Year: 2004)
(A) 8
(B) 9
(C) 10
(D) 11
Q23
If \(a_n=3n-5\), then the sequence is
(Exam: BITSAT Year: 2011)
(A) G.P.
(B) H.P.
(C) A.P.
(D) neither
Q24
The sum of the first 20 terms of the series \(2+4+6+\cdots\) is
(Exam: State Engg. Year: 2007)
(A) 400
(B) 410
(C) 420
(D) 440
Q25
If the ratio of the sum of first \(n\) terms of two A.P.s is independent of \(n\), then the ratio of their first terms equals the ratio of their
(Exam: IIT-JEE Year: 2002)
(A) common differences
(B) last terms
(C) number of terms
(D) sums
Q26
The value of \(1\cdot2+2\cdot3+3\cdot4+\cdots+n(n+1)\) is
(Exam: NEET Year: 2015)
(A) \(\frac{n(n+1)}{2}\)
(B) \(\frac{n(n+1)(n+2)}{3}\)
(C) \(\frac{n(n+1)(n+2)}{2}\)
(D) \(n(n+1)^2\)
Q27
If \(a,b,c\) are in H.P., then
(Exam: Olympiad Year: 2003)
(A) \(a,b,c\) are in A.P.
(B) \(\frac1a,\frac1b,\frac1c\) are in A.P.
(C) \(a^2=b^2+c^2\)
(D) \(b^2=ac\)
Q28
The common ratio of the G.P. whose terms are squares of the terms of an A.P. is
(Exam: IIT-JEE Year: 1993)
(A) constant
(B) zero
(C) not constant
(D) unity
Q29
If the sum of three numbers in G.P. is 21 and their product is 343, the middle term is
(Exam: NEET Year: 2010)
(A) 5
(B) 6
(C) 7
(D) 8
Q30
The sum of first \(n\) natural numbers is
(Exam: State Engg. Year: 2005)
(A) \(n^2\)
(B) \(\frac{n(n+1)}{2}\)
(C) \(\frac{n(n-1)}{2}\)
(D) \(\frac{n^2+n}{3}\)
Q31
If \(a_n=\frac{2n+1}{n+1}\), then \(\lim_{n\to\infty}a_n\) equals
(Exam: IIT-JEE Year: 2008)
(A) 1
(B) 2
(C) \(\infty\)
(D) 0
Q32
The sum of cubes of first \(n\) natural numbers equals
(Exam: NEET Year: 2017)
(A) \(\frac{n^2(n+1)^2}{4}\)
(B) \(\frac{n(n+1)(2n+1)}{6}\)
(C) \(n^3\)
(D) \(\frac{n(n+1)}{2}\)
Q33
If the first term of an A.P. is 1 and the sum of first \(n\) terms is \(n^2\), the common difference is
(Exam: IIT-JEE Year: 2000)
(A) 1
(B) 2
(C) 3
(D) 4
Q34
The sequence \(1,-\frac12,\frac14,-\frac18,\dots\) is
(Exam: BITSAT Year: 2013)
(A) A.P.
(B) G.P.
(C) H.P.
(D) neither
Q35
If the sum of infinite G.P. is finite, then
(Exam: IIT-JEE Year: 1992)
(A) \(r>1\)
(B) \(r<-1\)
(C) \(|r|<1\)
(D) \(r=1\)
Q36
The arithmetic mean of roots of \(x^2-6x+8=0\) is
(Exam: NEET Year: 2012)
(A) 3
(B) 4
(C) 5
(D) 6
Q37
If \(a,b,c\) are in A.P., then \(2b\) equals
(Exam: Olympiad Year: 2002)
(A) \(a+c\)
(B) \(ac\)
(C) \(a-c\)
(D) \(\frac{a+c}{2}\)
Q38
The sum of the series \(\frac12+\frac14+\frac18+\cdots+\frac1{2^n}\) is
(Exam: AIEEE Year: 2009)
(A) \(1-\frac1{2^n}\)
(B) \(1-\frac1{2^{n+1}}\)
(C) \(\frac1{2^{n-1}}\)
(D) \(2-\frac1{2^{n-1}}\)
Q39
If the \(n\)th term of an A.P. is \(7-3n\), then the common difference is
(Exam: State Engg. Year: 2010)
(A) 7
(B) -3
(C) 3
(D) -7
Q40
The geometric mean between 2 and 8 is
(Exam: NEET Year: 2009)
(A) 3
(B) 4
(C) 5
(D) 6
Q41
If the sum of first \(n\) terms of an A.P. is proportional to \(n^2\), then the first term is
(Exam: IIT-JEE Year: 1991)
(A) zero
(B) non-zero
(C) negative
(D) infinite
Q42
The series \(1+\frac13+\frac15+\cdots\) is
(Exam: Olympiad Year: 2006)
(A) convergent
(B) divergent
(C) finite
(D) G.P.
Q43
If the sum of three consecutive terms of an A.P. is 21 and their product is 231, the middle term is
(Exam: NEET Year: 2018)
(A) 6
(B) 7
(C) 8
(D) 9
Q44
The sum of the first 100 natural numbers is
(Exam: State Engg. Year: 2006)
(A) 4950
(B) 5000
(C) 5050
(D) 5100
Q45
If \(a,b,c\) are positive and in G.P., then
(Exam: IIT-JEE Year: 1990)
(A) \(b^2=ac\)
(B) \(2b=a+c\)
(C) \(b=\frac{a+c}{2}\)
(D) \(a^2=b+c\)
Q46
The sum of the series \(1-1+1-1+\cdots\) is
(Exam: Olympiad Year: 2008)
(A) 0
(B) 1
(C) \(\frac12\)
(D) divergent
Q47
If the common difference of an A.P. is zero, the sequence is
(Exam: NEET Year: 2019)
(A) increasing
(B) decreasing
(C) constant
(D) alternating
Q48
The value of \(\sum_{k=1}^n(2k-1)\) is
(Exam: AIEEE Year: 2007)
(A) \(n^2\)
(B) \(n(n+1)\)
(C) \(2n^2\)
(D) \(n(n-1)\)
Q49
If the ratio of successive terms of a sequence tends to 1, then the sequence is necessarily
(Exam: IIT-JEE Year: 2009)
(A) convergent
(B) divergent
(C) monotonic
(D) none
Q50
The sum of the first \(n\) terms of the sequence \(1,3,6,10,\dots\) is
(Exam: Olympiad Year: 2005)
(A) \(\frac{n(n+1)(n+2)}{6}\)
(B) \(\frac{n(n+1)}{2}\)
(C) \(\frac{n(n+1)(2n+1)}{6}\)
(D) \(\frac{n^2(n+1)}{2}\)

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