RD SHARMA Solutions for Class 11-science Maths Chapter 12 - Mathematical Induction

for all nN

Prove by the principle of mathematical induction

n³ - 7n + 3 is divisible by 3 for all n Î N

Prove by the principle of mathematical induction

1 + 2 + 2² +…. + 2ⁿ = 2^{n + 1} -1 for all n Î N

Prove by the principle of mathematical induction

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 2ⁿ for all n Î N.

A sequence a₁, a₂, a₃, …….. is defined by letting a₁ = 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 x₀, x₁, x₂, x₃, ……. is defined by letting x₀ = 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, a₁ and a₂, we have c (a₁ + a₂) = ca₁ + ca₂

Use this law and mathematical induction to prove that, for all natural numbers, n ³ 2, if c (a₁ + a₂ + …. + a_n) = ca₁ + ca₂ + …+ ca_n.

State the first principle of mathematical induction.

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

State the second principle of mathematical induction.