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Mathematical Induction Test - 3

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Mathematical Induction Test - 3
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  • Question 1
    1 / -0

    If P ( n ) = 2+4+6+………………..+2n , n ∈ N , then P ( k ) = k ( k + 1 ) + 2 ⇒ P ( k + 1 ) = ( k + 1 ) ( k +2 ) + 2 for all k ∈ N . So we can conclude that P ( n ) = n ( n + 1 ) +2 for

    Solution

    Because the statement is incomplete without the conclusion/ RHS

  • Question 2
    1 / -0

    The greatest positive integer, which divides (n+1)(n+2)(n+3)..................(n+r)∀n∈W

    Solution

    If n = 0 the given expression becomes 1.2.3.4........r = r! Also when n = 1 one more extra term will be there in the product  2.3.4........(r+1)which is also divisible by r!.

  • Question 3
    1 / -0
    Which of the following numbers will divide n3 + 3n2 + 5n + 3 for all n belonging to natural numbers?
  • Question 4
    1 / -0

    The greatest positive integer , which divides n ( n + 1 ) ( n + 2 ) ( n + 3 ) for all n ∈ N , is

    Solution

    If n = 1 then the statement becomes 1x2x3x4= 24 : the consecutive natural numbers when substituted will be multiples of 24.

  • Question 5
    1 / -0
    Directions: The figure (x) is followed by four alternative figures (a), (b), (c) and (d), such that figure (x) is embedded in one of them. Choose the alternative figure in which the figure (x) is embedded.

  • Question 6
    1 / -0

    (xn−yn)is divisible by ( x - y ) for 


    Solution

    Replacing n by 1,2,3... we get the expression with  the factor ( x - y ).Hence it is divisible by ( x - y ).

  • Question 7
    1 / -0
    How many circles will you get at 15th step, if the sequence follows the order as given in the following diagram?

  • Question 8
    1 / -0
    Which of the following will be true for every x  R?
  • Question 9
    1 / -0

    Let P(n) b e a statement 2n

    Solution

    Since for example n = 4 will give LHS as 16 and RHS as 4! = 1.2.3.4= 24

  • Question 10
    1 / -0
    If l = number of circles at Lth stage and m = number of circles at Mth stage, then

  • Question 11
    1 / -0

    Let P(n) be a statement and let P(n)⇒P(n+1) for all natural numbers n , then what will the nature of  P(n) ?

    Solution

    Since the statement is conditional statement ( if....then).

  • Question 12
    1 / -0
    How many circles will you get at 9th step, if the sequence follows the order as given in the following diagram?

  • Question 13
    1 / -0
    Identify the figure that completes the pattern.

  • Question 14
    1 / -0
    How many pieces of cube will constitute the big cube at 4th, 5th and 6th steps?

  • Question 15
    1 / -0
    If l = number of circles at Lth stage and m = number of circles at Mth stage, then
  • Question 16
    1 / -0
    Identify the figure that completes the pattern represented by 'X'.

  • Question 17
    1 / -0
    Directions: Select one figure from the answer figures which will continue the same series as given in the problem figures.

  • Question 18
    1 / -0

    The smallest +ve integer n , for which  n!<(n+1/2)n holds is

    Solution

    when n = 2 ,  L H S  :  n ! = 2 ! = 2x1 = 2    RHS: (3/2)2=9/4


    But if n = 1 , the inequation does not hold good.

  • Question 19
    1 / -0

    Let P ( n ) denote the statement n2+n is odd , It is seen that P(n)⇒P(n+1) , therefore P ( n ) is true for all

    Solution

    Since if n = 1 the answer is 2 which is even . also p ( 1 ) = p ( 2 ) is also not true since p(2 ) is 6 which is even

  • Question 20
    1 / -0
    What will be the sum of number of circles in 6th, 7th and 9th steps?

  • Question 21
    1 / -0

    The statement P ( n ) : “1 X 1! + 2 X 2! + 3 X 3! + …..+ n X n! = ( n + 1 ) !− 1 “ is 

    Solution

    Replace n = 1, 2 3  then LHS = RHS

  • Question 22
    1 / -0
    What will be the sum of number of circles in 6th, 7th and 8th steps?

  • Question 23
    1 / -0
    How many pieces of cube will constitute the big cube at 6th step?

  • Question 24
    1 / -0
    If l = number of circles at Lth stage and m = number of circles at Mth stage, then

  • Question 25
    1 / -0

    If x > -1 , then the statement (1+x)n>1+nx is true for

    Solution

    For example , if x = 0 and n > 1 , say n = 2 then the given inequality will become 12 > 1 , which is not true. Hence both the conditions on x and n are needed.

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