Mathematics - Sequences & Series

Question - 1

If the sum of the first 2n terms of the A.P. 2, 5, 8, .... is equal to the sum of the first n terms of the A.P. 57, 59, 61, ..... then n equal equals

  • A 10
  • B 12
  • C 11
  • D 13

Question - 2

If x, y, z (x>1, y>1, z>1) are three positive numbers in G.P. then \({1\over {1+1n\ x}}\)\({1\over {1+1n\ y}}\)\({1\over {1+1n\ z}}\) are in

  • A A.P.
  • B G.P.
  • C H.P.
  • D NONE OF THESE

Question - 3

Let an be the nth terms of a G.P. of positive integers. Let \(\overset { 100 }{ \underset { n=1 }{ \sum { { a }_{ 2n } } } } \quad =\quad \alpha \quad and\quad \overset { 100 }{ \underset { n=1 }{ \sum { { a }_{ 2n } } } } +1\quad =\quad \beta \) such that \(\alpha \quad \neq \quad \beta \) Thus common ratio of the G.P. is

  • A \(\frac { \alpha }{ \beta } \)
  • B \(\frac { \beta }{ \alpha } \)
  • C \({ \left( \frac { \alpha }{ \beta } \right) }^{ 1/2 }\)
  • D \({ \left( \frac { \beta }{ \alpha } \right) }^{ 1/2 }\)

Question - 4

Let T be the rth term of an A.P. for r = 1, 2, 3 .... If for some positive integers m, n \({ T }_{ m }\quad =\quad \frac { 1 }{ n } ,\quad { T }_{ n }\quad =\quad \frac { 1 }{ m } \)then Tmn equals

  • A \(1\over mn\)
  • B \({1\over m}+{1\over n}\)
  • C 1
  • D 0

Question - 5

Let the positive numbers a, b, c, d be in A.P. Then abc, abd, bcd are

  • A not in A.P./G.P./H.P.
  • B in A.P.
  • C in G.P.
  • D in H.P.

Question - 6

Consider an infinite series with first term a and common ratio r. If its sum is 4 and the second term is \(3\over 4\)then

  • A \(a\quad =\quad \frac { 7 }{ 4 } ,\quad r\quad =\quad \frac { 3 }{ 7 } \)
  • B \(a\quad =\quad 2,\quad r\quad =\quad \frac { 3 }{ 8 } \)
  • C \(a\quad =\quad \frac { 3 }{ 2 } ,\quad r\quad =\quad \frac { 1 }{ 2 } \)
  • D \(a\quad =\quad 3,\quad r\quad =\quad \frac { 1 }{ 4 } \)

Question - 7

If a, b , c are positive numbers, the least value of \(\frac { a+b }{ c } +\frac { b+c }{ a } +\frac { c+a }{ b } \)is

  • A 6
  • B 4
  • C 3
  • D NONE OF THESE

Question - 8

If a, b, c are positive numbers then value of \((a+b+c)\left( \frac { 1 }{ a } +\frac { 1 }{ b } +\frac { 1 }{ c } \right) \)is

  • A \(\ge 10\)
  • B \(\ge 9\)
  • C \(\ge 12\)
  • D NONE OF THESE

Question - 9

If the 9th term of A.P. is zero, then ratio of 29th term to 19th term is

  • A 2 : 1
  • B 3 : 1
  • C 9 : 1
  • D NONE OF THESE

Question - 10

The sum of the first m terms of an A.P. is the same as the sum of the n terms, where \(m\quad \neq \quad n\)Then the sum of the first (m+n) terms is

  • A m
  • B n
  • C 0
  • D NONE OF THESE
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