## Chapter 12 – Mathematical Induction Exercise Ex. 12.1

## Chapter 12 – Mathematical Induction Exercise Ex. 12.2

Prove by the principle of mathematical induction

n^{3} – 7n + 3 is divisible by 3 for all n Î N

Prove by the principle of mathematical induction

1 + 2 + 2^{2} +…. + 2^{n} = 2^{n + 1}

-1 for all n Î N

Prove by the principle of mathematical induction

for all n_{}N

Prove that

cos a + cos (a + b) + cos (a + 2b) + …..+ cos (a + (n – 1)b)

Prove that the number of subsets of a set containing n

distinct elements is 2^{n} for all n Î N.

A sequence a_{1}, a_{2}, a_{3}, …….. is defined by

letting a_{1} = 3 and a_{k} = 7 a_{k-1}

for all natural numbers k ³ 2. Show that a_{n} = 3.7^{n-1} for all

n Î N.

A sequence x_{0}, x_{1}, x_{2},

x_{3}, ……. is defined by letting x_{0}

= 5 and x_{k} = 4 + x_{k}_{ -1} for all natural number k. show that x_{n}

= 5 + 4n for all n Î N using mathematical induction.

Using principle of mathematical induction prove that

The distributive law from algebra states that for all

real numbers c, a_{1} and a_{2}, we have c (a_{1} + a_{2})

= ca_{1} + ca_{2}

Use this law and mathematical induction to prove that,

for all natural numbers, n ³ 2, if c (a_{1} + a_{2} + …. + a_{n}) = ca_{1} + ca_{2} + …+ ca_{n}.

## Chapter 12 – Mathematical Induction Exercise Ex. 12VSAQ

State the first principle of mathematical induction.

Write the set of value of n for which the statement P(n):2n

N-{1,2,3} Where N is the set of all natural numbers

State the second principle of mathematical induction.