Prove the following by the principle of mathematical induction:

Answer:

… … … … … i)

Now substituting the from equation i)

Hence by the principle of mathematical induction

Answer:

… … … … … i)

Now substituting the from equation i)

Hence by the principle of mathematical induction

Answer:

… … … … … i)

Now substituting the from equation i)

Hence by the principle of mathematical induction

Answer:

… … … … … i)

Now substituting the from equation i)

Hence by the principle of mathematical induction

i.e., the sum of firs odd numbers is

Answer:

… … … … … i)

Now substituting the from equation i)

Hence by the principle of mathematical induction

Answer:

… … … … … i)

Hence by the principle of mathematical induction

Answer:

… … … … … i)

Hence by the principle of mathematical induction

Answer:

… … … … … i)

Answer:

… … … … … i)

Hence by the principle of mathematical induction

Answer:

… … … … … i)

Hence by the principle of mathematical induction

Answer:

… … … … … i)

Hence by the principle of mathematical induction

Answer:

… … … … … i)

Hence by the principle of mathematical induction

Answer:

… … … … … i)

Hence by the principle of mathematical induction

Answer:

… … … … … i)

Hence by the principle of mathematical induction

Answer:

… … … … … i)

Hence by the principle of mathematical induction

Answer:

… … … … … i)

Hence by the principle of mathematical induction

Answer:

… … … … … i)

Hence by the principle of mathematical induction

Answer:

… … … … … i)

Hence by the principle of mathematical induction

is divisible by for all

Answer:

is divisible by for all

which is divisible by 24

Hence

is divisible by for all

… … … … … i)

where

Hence by the principle of mathematical induction

is divisible by for all

is divisible by for all

Answer:

is divisible by for all

which is divisible by

Hence

is divisible by for all

… … … … … i)

where

Hence by the principle of mathematical induction

is divisible by for all

is divisible by for all

Answer:

is divisible by for all

which is divisible by

Hence

is divisible by for all

… … … … … i)

where

Hence by the principle of mathematical induction

is divisible by for all

is divisible by for all

Answer:

is divisible by for all

which is divisible by

Hence

is divisible by for all

… … … … … i)

where

Hence by the principle of mathematical induction

is divisible by for all

for all

Answer:

for all

RHS Therefore LHS = RHS

Hence

for all

Hence by the principle of mathematical induction

for all

is a multiple of for all

Answer:

is a multiple of for all

which is divisible by

Hence

is divisible by for all

… … … … … i)

where

Hence by the principle of mathematical induction

is a multiple of for all

is divisible by for all

Answer:

is divisible by for all

which is divisible by

Hence

is divisible by for all

… … … … … i)

where

Hence by the principle of mathematical induction

is divisible by for all