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\(n(n+1) (n+5) \) is multiple of _______ ,\( \forall n\in N \)
For , is a multiple of .
For , is a multiple of .
For is true.
For , is multiple of
is true.
Find the number of shots arranged in a complete pyramid the base of which is an equilateral triangle, each side containing \( n \) shots.
The number of shots in the lower layer
.
We obtain the number of shots in 2nd, 3rd,โฆlayer by written
,… respectively.
So, total shots .
\((n^2+n) \) is _ ,\( \forall n\in N \)
For even.
For even.
For even.
For
, is even.
is true.
, is even, .
For all \(n\in N \), \(3 \times 5^{2n+1}+ 2^{3n+1} \) is divisible by__________.
For , is divisible by .
For , is divisible by .
For is true.
For
is divisible by .
is true.
is true, .
\(f(n)= 12^n +25^{n1} \) is divisible by ____________.
For
For
For is divisible by .
For
is divisible by .
is true.
is true, .
\(f(n)= 11^{n+2}+12^{2n+1} \) is divisible by______________.
For , is divisible by .
For , is divisible by .
For
, is divisible by .
is true.
is true, .
\(f(n)= 2 \times 7^n +3\times5^n5 \) is divisible by ___________.
For
For
For is divisible by .
For
Since and are always odd, then is always even.
, is divisible by .
is true.
is true,.
For \( \forall n\in N \),\( f(n)=41^n14^n \) is divisible by___________.
For
For is true.
For
is a multiple of .
, is divisible by .
is true.
is true, .
\( 2^{3n}1 \) is divisible by __________.
, For
For where is true.
For
is divisible by .
is true.
is divisible by .
\( 4^n+15n1 \) is divisible by ____________.
For ,is divisible by .
For , is divisible by .
For is true.
For
, is divisible by .
is true.
is divisible by .
\( x^ny^n \) is divisible by \( (x+y) \) when _____________.
For , is divisible by .
For is true.
For
is true.
is divisible by when is even.
\(f(n)=n^3+ (n+1)^3+(n+2)^3 \) is divisible by ___________.
For , is divisible by .
For is true.
For
is divisible by .
is divisible by .
For \(n\in N, 3^{3n} 26n1 \),is divisible by __________.
is true.
is true by induction hypothesis.
is divisible by .
For all \(n\in N, n(n+1)(2n+1) \) is divisible by _________.
For
For , is divisible by .
For is true.
For
, is divisible by .
is true.
is true, .
\( f(n)=2^{3n}7^n1, \forall n\in N \) is divisible by ____________.
For
For
For is true.
By induction hypothesis it is true for .
is true.
is true, .
For is true.
The number of terms in the expansion of \((n+y+z)^n \) is ___________.
Number of terms
For
Hence number of terms are
A student asked to prove a statement \( P(n) \) by method of induction. He proved \( P(k+1) \) is true whenever \( P(k) \) is true \( \forall k \geq 5,K \in N \) and \( P(5) \) is true. On this basis he could conclude that \( P(n) \) is true_________.
If is true and truth of implies that is also true, we can conclude that is true , where .
Hence, we can conclude that is true .
\( 1\cdot 2\cdot 3 +2\cdot 3\cdot 4 + 3\cdot 4 \cdot 5+… \) up to \( n \) terms is equal to ______________.
nth term , where
, if nth term .
, if nth term .
, if nth term .
Hence the required sum
Consider the statement \( P(n) : “n^2 \geq 100” \), hence \( P(n)\Rightarrow P(n+1) \), for some \( n \), then_________.
For
\(P(n): n^2n+41 \) is a Prime Number , then _________.
, is a prime number.
, is prime number.
, is a prime number.
, is not a prime number.
is not true for is a prime number.
\((n+3)^2 > 2^{n+3} \) is true for ________________.
For
, it is true.
, it is false
, it is false
is false for i.e., .
no i.e., satisfy the above inequation.
\(3^{2n}1 \) is divisible by ____________.
For
For , is divisible by.
for is true.
For
, is divisible by .
is true.
is divisible by .
\(a^{2n1}+b^{2n1} \) is divisible by ____________.
For
For
For is true
For
is true
is true
\(P(n): 2^n(n1)! \) < \(n^n\) is true if
is false
is false
is true for
\(3^{3n}2n+1 \) is divisible by ____________.
is true for
is true for by induction hypothesis
is divisible by .