Science (English Medium) Class 11 - CBSE Question Bank Solutions

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.

Prove the statement by using the Principle of Mathematical Induction:

2 + 4 + 6 + ... + 2n = n² + n for all natural numbers n.

Prove the statement by using the Principle of Mathematical Induction:

1 + 2 + 2² + ... + 2ⁿ = 2ⁿ⁺¹ – 1 for all natural numbers n.

Prove the statement by using the Principle of Mathematical Induction:

1 + 5 + 9 + ... + (4n – 3) = n(2n – 1) for all natural numbers n.

A sequence a₁, a₂, a₃ ... is defined by letting a₁ = 3 and a_k = 7a_{k – 1} for all natural numbers k ≥ 2. Show that a_n = 3.7^n–1 for all natural numbers.

A sequence b₀, b₁, b₂ ... is defined by letting b₀ = 5 and b_k = 4 + b_{k – 1} for all natural numbers k. Show that b_n = 5 + 4n for all natural number n using mathematical induction.

A sequence d₁, d₂, d₃ ... is defined by letting d₁ = 2 and d_k = `(d_(k - 1))/"k"` for all natural numbers, k ≥ 2. Show that d_n = `2/(n!)` for all n ∈ N.

Prove that for all n ∈ N.
cos α + cos(α + β) + cos(α + 2β) + ... + cos(α + (n – 1)β) = `(cos(alpha + ((n - 1)/2)beta)sin((nbeta)/2))/(sin  beta/2)`.

Prove that, cosθ cos2θ cos2²θ ... cos2^n–1θ = `(sin 2^n theta)/(2^n sin theta)`, for all n ∈ N.

Prove that, sinθ + sin2θ + sin3θ + ... + sinnθ = `((sin ntheta)/2 sin  ((n + 1))/2 theta)/(sin  theta/2)`, for all n ∈ N.

Show that `n^5/5 + n^3/3 + (7n)/15` is a natural number for all n ∈ N.