Arts (English Medium) इयत्ता ११ - CBSE Question Bank Solutions

The distributive law from algebra says 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 ≥ 2, if c, a₁, a₂, ..., a_n are any real numbers, then c(a₁ + a₂ + ... + a_n) = ca₁ + ca₂ + ... + ca_n.

Prove by induction that for all natural number n sinα + sin(α + β) + sin(α + 2β)+ ... + sin(α + (n – 1)β) = `(sin (alpha + (n - 1)/2 beta)sin((nbeta)/2))/(sin(beta/2))`

Prove by the Principle of Mathematical Induction that 1 × 1! + 2 × 2! + 3 × 3! + ... + n × n! = (n + 1)! – 1 for all natural numbers n.

Show by the Principle of Mathematical Induction that the sum S_n of the n term of the series 1² + 2 × 2² + 3² + 2 × 4² + 5² + 2 × 6² ... is given by

S_n = `{{:((n(n + 1)^2)/2",",  "if n is even"),((n^2(n + 1))/2",",  "if n is odd"):}`

Let P(n): “2ⁿ < (1 × 2 × 3 × ... × n)”. Then the smallest positive integer for which P(n) is true is ______.

A student was asked to prove a statement P(n) by induction. He proved that P(k + 1) is true whenever P(k) is true for all k > 5 ∈ N and also that P(5) is true. On the basis of this he could conclude that P(n) is true ______.

State whether the following proof (by mathematical induction) is true or false for the statement.

P(n): 1² + 2² + ... + n² = `(n(n + 1) (2n + 1))/6`

Proof By the Principle of Mathematical induction, P(n) is true for n = 1,

1² = 1 = `(1(1 + 1)(2*1 + 1))/6`. Again for some k ≥ 1, k² = `(k(k + 1)(2k + 1))/6`. Now we prove that

(k + 1)² = `((k + 1)((k + 1) + 1)(2(k + 1) + 1))/6`

Give an example of a statement P(n) which is true for all n ≥ 4 but P(1), P(2) and P(3) are not true. Justify your answer

Give an example of a statement P(n) which is true for all n. Justify your answer. 

Prove the statement by using the Principle of Mathematical Induction:

4ⁿ – 1 is divisible by 3, for each natural number n.

Prove the statement by using the Principle of Mathematical Induction:

2³ⁿ – 1 is divisible by 7, for all natural numbers n.

Prove the statement by using the Principle of Mathematical Induction:

n³ – 7n + 3 is divisible by 3, for all natural numbers n.

Prove the statement by using the Principle of Mathematical Induction:

3²ⁿ – 1 is divisible by 8, for all natural numbers n.

Prove the statement by using the Principle of Mathematical Induction:

For any natural number n, 7ⁿ – 2ⁿ is divisible by 5.

Prove the statement by using the Principle of Mathematical Induction:

For any natural number n, xⁿ – yⁿ is divisible by x – y, where x and y are any integers with x ≠ y.

Prove the statement by using the Principle of Mathematical Induction:

n³ – n is divisible by 6, for each natural number n ≥ 2.

Prove the statement by using the Principle of Mathematical Induction:

n(n² + 5) is divisible by 6, for each natural number n.

Prove the statement by using the Principle of Mathematical Induction:

n² < 2ⁿ for all natural numbers n ≥ 5.

Prove the statement by using the Principle of Mathematical Induction:

2n < (n + 2)! for all natural number n.

Prove the statement by using the Principle of Mathematical Induction:

`sqrt(n) < 1/sqrt(1) + 1/sqrt(2) + ... + 1/sqrt(n)`, for all natural numbers n ≥ 2.