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