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\(a^n b^n \) is divisible by ____________., (where \( a \) & \( b \) are distinct.)
For is true.
For
, is divisible by .
is true.
, is divisible by .
For all \( n \geq 2, n^2 (n^41) \) is divisible by ___________.
For is true.
For is true by induction hypothesis.
is divisible .
For each \( n \in N \), the correct statement is ___________.
is true.
, is true.
is true.
is true by induction hypothesis.
is true.
For each \( n \in N , 10^{n2} > 81n\) if __________.
is false.
is false.
is false.
is false.
is true.
is true.
For is true, .
For is true by induction hypothesis.
is true if .
If \(a_n=2^{2^n}+1 \), for \(n > 1, (n=2a) a \in N \), the last digit of \( a_n \) is___________.
For
For
For
The last digit of , if is even.
\(1+3+3^2+…+3^{n1} \) is ___________.
For is true.
For
is true.
is true.
\(1+\dfrac{1}{(1+2)}+\dfrac{1}{(1+2+3)}+…+\dfrac{1}{(1+2+…+n)} \) is equal to ____________.
For is true.
For
is true by induction hypothesis.
.
\(f(n)=1\cdot3+2\cdot3^2+…+n \cdot3^n \) is equal to _____________.
For is true.
is true by induction hypothesis.
is true, .
\( f(n)=1\cdot 2+2\cdot 2^2+3\cdot 2^2+…+n\cdot2^n \) is equal to __________.
For is true.
is true by induction hypothesis.
.
\(f(n)=\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+…+\dfrac{1}{2^n} \) is equal to _______________.
For is true.
For
, is true.
is true.
is true.
\( f(n)= \dfrac{1}{2\cdot 5}+ \dfrac{1}{5\cdot 8}+ \dfrac{1}{8\cdot 11}+…+ \dfrac{1}{(3n1)(3n+2)} \) is __________.
For is true.
is true by induction hypothesis.
.
\(f(n)= a+an+an^2+…+an^{n1} \) is equal to ___________.
For is true.
is true by induction hypothesis.
.
\( f(n)= \left ( 1+\dfrac{3}{1} \right )\left ( 1+\dfrac{5}{4} \right )\left ( 1+\dfrac{7}{9} \right )…\left ( 1+\dfrac{(2n+1)}{n^2} \right ) \) is equal to________.
For is true.
is true by induction hypothesis.
.
\( f(n)= \left ( 1+\dfrac{1}{1} \right )\left ( 1+\dfrac{1}{2} \right )\left ( 1+\dfrac{1}{3} \right )…\left ( 1+\dfrac{1}{n} \right ) \) is equal to________.
For is true.
, is true.
.
\( f(n)=1^2+3^2+5^2+…+(2n1)^2 \) is equal to ___________.
For is true.
is true by induction hypothesis.
.
\( f(n)=\dfrac{1}{1\cdot4}+\dfrac{1}{4\cdot7}+\dfrac{1}{7\cdot10}+…+\dfrac{1}{(3n2)(3n+1)} \) is _________.
For is true.
is true by induction hypothesis.
.
\( f(n)=x^{2n}y^{2n}\) is divisible by___________.
For is true.
For
, is divisible by
is true.
is true, .
\(f(n):1+2+3+…+n <\dfrac{1}{8}(2n+1)^2 \), if________.
, is true.
, it is true.
For is true.
For is true by induction hypothesis.
, is true for all .
\(f(n): (2n+7) < (n+3)^2 \) is true if _______.
, it is true.
, it is true.
For is true.
By induction hypothesis we will get that for is true.
is true, .
\( 3^{2n+2}8n9 \) is divisible by _________.
For , is true.
for is true by induction hypothesis.
is divisible by .
\(1^2+2^2+…+n^2 >\dfrac{n^3}{3} \) is true if ___________.
, is true.
, is true.
, is true.
For , is true.
By induction hypothesis is true.
.
\(f(n):2^n > n \) is true if _________.
, is true.
, is true.
For , is true.
For is true by induction hypothesis.
\(f(n): (1+x)^n \geq (1+nx) \), where \( n > 1 \) is true if______________.
, is true.
is true.
For is true.
For is true by induction hypothesis.
\(f(n)=11^n2^n \) is divisble by_______.
For is true.
For ,
, true.
is divisible by .
\(P(n):\dfrac{4^n}{n+1} < \dfrac{(2n)!}{(n!)^2} \), then \(P(n) \) is true for_______.
, is not true.
, is true.
is true for .