Question 26: is divisible by
for all
Answer:
Let
is divisible by
for all
For , L.H.S
which is divisible by
Hence is true for
Let is true for
is divisible by
for all
… … … … … i)
Now we have to show that is true for
We know that is divisible by
. Hence
is true for
Hence by the principle of mathematical induction
is divisible by
for all
Question 27: is divisible by
for all
Answer:
Let
is divisible by
for all
For , L.H.S
which is divisible by
Hence is true for
Let is true for
is divisible by
for all
… … … … … i)
Now we have to show that is true for
where
is true for
Hence by the principle of mathematical induction
is divisible by
for all
Question 28: for all
Answer:
Let
for all
For , L.H.S
R.H.S
Therefore LHS = RHS
Hence is true for
Let is true for
for all
… … i)
Now we have to show that is true for
is true for
Hence by the principle of mathematical induction
for all
Question 29: is divisible by
for all
Answer:
Let
is divisible by
for all
For , L.H.S
which is divisible by
Hence is true for
Let is true for
is divisible by
for all
… … … … … i)
Now we have to show that is true for
where
is true for
Hence by the principle of mathematical induction
is divisible by
for all
Question 30: for all
Answer:
Let
for all
For , L.H.S
R.H.S
Therefore LHS = RHS
Hence is true for
Let is true for
for all
… … … … … i)
Now we have to show that is true for
is true for
Hence by the principle of mathematical induction
for all
Question 31:
Answer:
Let
For , L.H.S
R.H.S
Therefore LHS = RHS
Hence is true for
Let is true for
… … … … … i)
Now we have to show that is true for
is true for
Hence by the principle of mathematical induction
Question 32: Prove that
is a positive integer got all
Answer:
Let
is a positive integer got all
For , L.H.S
which is positive.
Hence is true for
Let is true for
is a positive integer got all
… … … … … i)
Now we have to show that is true for
is true for
Hence by the principle of mathematical induction
is a positive integer got all
Question 33: Prove that
is a positive integer got all
Answer:
Let
is a positive integer got all
For ,
L.H.S
1 is an integer. Hence is true for
Let is true for
is a positive integer got all
Now we have to show that is true for
Which is an integer. is true for
Hence by the principle of mathematical induction
is a positive integer got all
Question 34: Prove that
for all and
Answer:
Let
for all and
For , L.H.S
R.H.S
Therefore LHS = RHS
Hence is true for
Let is true for
… i)
Now we have to show that is true for
is true for
Hence by the principle of mathematical induction
Question 35: Prove that
for all natural numbers,
Answer:
Let
for all natural numbers,
For , L.H.S
R.H.S
Therefore LHS = RHS
Hence is true for
Let is true for
for all natural numbers,
… … i)
Now we have to show that is true for
is true for
Hence by the principle of mathematical induction
for all natural numbers,
Question 36: Prove that
for all
Answer:
Let
for all
For ,
Hence is true for
Let is true for
for all
… … … … … i)
Now we have to show that is true for
Since
Since
is true for
Hence by the principle of mathematical induction
for all
Question 37:
for all
Answer:
Let
for all
For ,
Hence is true for
Let is true for
for all
… … … … … i)
Now we have to show that is true for
is true for
Hence by the principle of mathematical induction
for all
Question 38: Prove that is divisible by
for all
Answer:
Let
is divisible by
for all
For which is divisible by
Hence is true for
Let is true for
is divisible by
for all
… … … … i)
Now we have to show that is true for
where
is true for
Hence by the principle of mathematical induction
is divisible by
for all
Question 39: Prove that
for all
Answer:
Let
for all
For
latex P(1) $ is true for
Let is true for
for all
… … … i)
Now we have to show that is true for
is true for
Hence by the principle of mathematical induction
for all
Question 40: Prove that
Answer:
Let
For
RHS
Therefore LHS = RHS
Hence is true for
Let is true for
… i)
Now we have to show that is true for
is true for
Hence by the principle of mathematical induction
Question 41: Prove that
, for all natural numbers
Answer:
Let
, for all natural numbers
For , L.H.S
R.H.S
Hence is true for
Let is true for
, for all natural numbers
… … … … … i)
Now we have to show that is true for
is true for
Hence by the principle of mathematical induction
, for all natural numbers
Question 42: Give
and
for , where
Prove that
Answer:
Let
For LHS
RHS
Therefore LHS = RHS
Hence is true for
Let is true for
is true for
… … … … … i)
Now we have to show that is true for
LHS
RHS
Therefore LHS = RHS
is true for
Hence by the principle of mathematical induction
Question 43: Let be the statement:
. If
is true, show that
is true. Do you conclude that
is true for all
?
Answer:
Let
for all
For , LHS
RHS
Therefore LHS RHS
Hence is NOT true for
Therefore is NOT true for all
Question 44: Show that by the principle of mathematical induction that the sum of the
terms of the series
is given by
Answer:
Let
of the
terms of the series
is given by
For ,
L.H.S
R.H.S
Therefore LHS = RHS
Hence is true for
For ,
L.H.S
R.H.S
Therefore LHS = RHS
Hence is true for
Let is true for
of the
terms of the series
is given by
… … … … … i)
Now we have to show that is true for
If is even,
is odd
If is odd,
is even
is true for
Hence by the principle of mathematical induction
of the
terms of the series
is given by
Question 45: Prove that the number of subsets of a set containing distinct elements is
for all
Answer:
Let P(n) : The number of subsets of a set containing distinct elements is
for all
For LHS
number of subsets of a set containing only
element
and the set itself
RHS
.
Hence is true for
Let is true for
The number of subsets of a set containing distinct elements is
for all
Now we have to show that is true for
Let
Now
Using i) we can say that has
subset and
as
subset.
has
subsets.
is true for
Hence by the principle of mathematical induction The number of subsets of a set containing distinct elements is
for all
Question 46: A sequence is defined by letting
and
for all natural numbers
. Show that
for all
.
Answer:
Given a sequence is defined by letting
and
for all natural numbers
Let
for all
For , L.H.S
Hence is true for
Let is true for
for all
… … i)
Now we have to show that is true for
is true for
Hence by the principle of mathematical induction for all
Question 47: A sequence is defined by letting
and
for all natural numbers
. Show that
for all
.
Answer:
Given a sequence is defined by letting
and
for all natural numbers
.
Let
for all
.
For , L.H.S
Hence is true for
Let is true for
for all
… … i)
Now we have to show that is true for
is true for
Hence by the principle of mathematical induction
for all
Question 48: A sequence is defined by letting
and
for all natural numbers
. Show that
for all
using mathematical induction.
Answer:
Given A sequence is defined by letting
and
for all natural numbers
.
Let
for all
For , L.H.S
Hence is true for
Let is true for
for all
… … i)
Now we have to show that is true for
is true for
Hence by the principle of mathematical induction
for all
Question 49: Using the principle of mathematical induction prove that
for all natural numbers
.
Answer:
Let
for all natural numbers
For , L.H.S
R.H.S
Therefore LHS < RHS
Hence is true for
Let is true for
for all natural numbers
… … i)
Now we have to show that is true for
LHS
RHS
Since
Therefore LHS RHS
is true for
Hence by the principle of mathematical induction
for all natural numbers
Question 50: The distributive law from algebra states that for all real numbers and
we have
. Use this law and mathematical induction to prove that, for all natural numbers,
, if
are any real numbers, then
Answer:
For all real numbers , and
To prove that, for all natural numbers, , if
are any real numbers, then
Let
for all natural numbers
, and
For , LHS
, RHS
LHS
RHS
Hence is true for
Let is true for
therefore
for all natural numbers
, and
For
Therefore is true for
Hence by the principle of mathematical induction
for all natural numbers
, and