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\( \forall n \in N, \cos\theta \cdot \cos2\theta \cdot \cos 4\theta …\cos2^{n1}\theta \) is equal to________.
For
is true by mathematical induction.
Hence
\( 5^{2n+1+}+3^{n+2}\cdot2^{n1} \)
For is true.
For is true by induction hypothesis.
is divisible by .
Product of three consecutive natural number is divisible by_______.
Let be three consecutive numbers.
For
For
For is divisible by .
for ,
is true.
Hence product of three consecutive numbers is divisible by .
The smallest positive integer \( n \) which \(n! <\left \{ \dfrac{(n+1)}{2} \right \}^n \) holds, is ________.
, is not true.
, is true.
is true.
For is true.
By induction hypothesis , is also true.
is true for .
The greatest positive number, which divides \( (n+2)(n+3)(n+4)(n+5)(n+6) \) is____________.
For is true.
For is divisible by by induction hypothesis.
is divisible by the greatest positive integer
\(f(n)= x(x^{n1}n\alpha ^{n1})+\alpha ^n(n1) \) is divisible by ________.
For is true.
For is divisible by by induction hypothesis.
is divisible by .
\( f(n)=2\cdot4^{2n+1}+3^{3n+1} \) is divisible by ___________.
For is true.
For is divisible by by induction hypothesis.
is divisible by .
Let \( f(n): n^2+n+1 \) is an odd integer. If \( f(k) \) is true \( \implies f(k+1) \) is true. Therefore, \( f(n) \) is true for_________.
, is an odd number.
, is an odd number.
is true is true.
is true .
\(3^n <n!, n \in N \) is true for__________.
, is not true.
, is not true.
, is not true.
, is not true.
, is true.
For is true.
is true by induction hypothesis.
is true for .
\( \forall n \in N , f(n) \) is a statement such that if \( f(k) \) is true \(\implies f(k+1) \) is true for \( K \in N \), then \( f(n) \) is true for ________.
is a statement
if is true is true, . But it can not be confimed that the value of may varies according to the statement.
\(f(n)=1+3+5+…+(2^n1)3+n^2 \), then which of the following is true?
The statement is true when is true is true we can’t say that is true. If is true, is not true then the statement can’t be prove by induction method.
If \(f(n)=1+3+5+…+(2n1)=n^2 \) is true for ______.
is true.
is true by induction hypothesis.
is true for all .
If \(A=\begin{bmatrix}
1 &0 \\
1&1
\end{bmatrix} \) & \( I=\begin{bmatrix}
1 &0 \\
0&1
\end{bmatrix} \), which of the following holds \( \forall n \in N \).
,
For is true.
is true by induction.
\(10^n+3(4^{n+2})+5 \) is divisible by_______.
For is true.
is true by induction hypothesis.
is divisible by .
\((3+5^{\frac{1}{2}})^n +(35^{\frac{1}{2}})^n \) is equal to______.
For is true.
is true.
.