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What is the base case for the inequality \(7^n> 3^n\), where \(n =3\)
By Mathematical Induction, we have as a base case and it is true for .
In the Principle of Mathematical Induction, which of the following steps is mandatory?
The hypothesis of step is a must for Mathematical Induction i.e., the statements is true for , where , which is called Induction Hypothesis.
For \(m =1,2โฆ,4m+2 \) is a multiple of _____
For , is divisible by .
, is divisible by .
Since are divisbile by .
, is divisible by .
For ,
, is divisible by
or is a multiple of
.
Hence is a multiple of , for
For all \(n \geq {1} , f(n) =1^2+2^2+3^2+…+n^2 \) is equal to _____
For
For
For
It is true for
For
is true.
is true by induction.
\( f(n)=1+3+5+f+…+(2n1), \forall \ n\in \ N\) is equal to _____
For
For
For
It is true for i.e.,
For
is true.
is true.
Hence
For any \( n\in \ N \), which of the following is true?
For
For
For
is true
is true
is true
For \( n\geq \ 1,\dfrac{1}{1\times 2}+\dfrac{1}{2\times 3}+\dfrac{1}{3\times 4}+…+ \dfrac{1}{n(n+1)}\) is equal to _______
To show that by induction
For
For is true
For n = K+1
is true
is true
According to Principle of Induction if \( P(k+2) = m^{K+2}+5\) is true, then _______ must be true.
is true.
Hence is true.
is definitely true.
Which of the following is the base case for \( 4^{n+1}>(n+1)^2 \) where \(n=2 \)?
By Mathematical Induction, for n= 2 the basecase of the equation,
What is the induction hypothesis for the inequality \(m! >2^m ? \) where \( m\geq 4 \)
By induction, we assume that is true.
For holds where .
For \(n\in \ N , 1+2+3+…+n \) is equal to?
For
For
For is true.
then for
is true.
Hence is true, for all
For all \(n\in \ N, 7^n 3^n \) is divisible by _______.
For divisible by .
For divisible by .
For is true.
For n=k+1, is divisible by .
is true, for some
is true, .
\( f(n)=1^3+2^3+3^3+…+n^3 \) is equal to
For
For
For
For is true.
is true.
is true, for all
\( f(n)=\left \{ 1\dfrac{1}{2} \right \}\left \{ 1\dfrac{1}{3} \right \}\left \{ 1\dfrac{1}{4} \right \}…\left \{ 1\dfrac{1}{n+1} \right \}\) is equal to..
For
For
For
Hence for n = k+1
is true.
is true.
For any natural number \( n,7^n2^n \) is divisible by_______
For , is divisible by
For , is divisible by
For , is divisible by
For , is divisible by
is divisible by
is true.
is true, .
\(\dfrac{1}{1 \times 2 \times 3}+\dfrac{1}{2 \times 3\times 4}+…+\dfrac{1}{n(n+1) (n+2)}= \)
For
For is true
For is true by induction
\(10^{2n1}+1\) is divisible by _______, \( \forall \ n\in \ N \)
For , divisible by
For , divisible by
For , divisible by
For
, is divisible by
is true, .
\(f(n)=7^{2n}48n1 \) is divisible by _______, \( \forall \ n\in N \)
For , is divisible by .
For , is true i.e.,
For is true by induction hypothesis.
is divisible by 2304.
\( \left ( \dfrac{1}{3\times 5} \right )+\left ( \dfrac{1}{5\times 7} \right )+\left ( \dfrac{1}{7\times 9} \right )+…+\left \{ \dfrac{1}{(2n+1)(2n+3)} \right \}= \)
is true.
For
is true
is true for all .
The nth term of the series \(3+7+13+21+… \) is
Let .
So, nth term is