HSC Science (Computer Science) इयत्ता ११ वी - Maharashtra State Board Question Bank Solutions

Prove by method of induction, for all n ∈ N:

5 + 5² + 5³ + .... + 5ⁿ = `5/4(5^"n" - 1)`

Prove by method of induction, for all n ∈ N:

(cos θ + i sin θ)ⁿ = cos (nθ) + i sin (nθ)

Prove by method of induction, for all n ∈ N:

Given that t_n+1 = 5t_n + 4, t₁ = 4, prove that t_n = 5ⁿ − 1

Prove by method of induction, for all n ∈ N:

`[(1, 2),(0, 1)]^"n" = [(1, 2"n"),(0, 1)]` ∀ n ∈ N

Answer the following:

Prove, by method of induction, for all n ∈ N

8 + 17 + 26 + … + (9n – 1) = `"n"/2(9"n" + 7)`

Answer the following:

Prove, by method of induction, for all n ∈ N

1² + 4² + 7² + ... + (3n − 2)² = `"n"/2 (6"n"^2 - 3"n" - 1)`

Answer the following:

Prove, by method of induction, for all n ∈ N

2 + 3.2 + 4.2² + ... + (n + 1)2^n–1 = n.2ⁿ 

Answer the following:

Prove, by method of induction, for all n ∈ N

`1/(3.4.5) + 2/(4.5.6) + 3/(5.6.7) + ... + "n"/(("n" + 2)("n" + 3)("n" + 4)) = ("n"("n" + 1))/(6("n" + 3)("n" + 4))`

Answer the following:

Given that t_n+1 = 5t_n − 8, t₁ = 3, prove by method of induction that t_n = 5ⁿ⁻¹ + 2

Answer the following:

Prove by method of induction

`[(3, -4),(1, -1)]^"n" = [(2"n" + 1, -4"n"),("n", -2"n" + 1)], ∀  "n" ∈ "N"`

Answer the following:

Prove by method of induction log_a xⁿ = n log_ax, x > 0, n ∈ N

Answer the following:

Prove by method of induction 15^2n–1 + 1 is divisible by 16, for all n ∈ N.

Answer the following:

Prove by method of induction 5²ⁿ − 2²ⁿ is divisible by 3, for all n ∈ N

Describe the following set in Roster form

A = {x/x is a letter of the word 'MOVEMENT'}

Describe the following set in Roster form

B = `{x//x  "is an integer", -3/2 < x < 9/2}`

Describe the following set in Roster form

C = {x/x = 2n + 1, n ∈ N}

Describe the following set in Set-Builder form

{0}

Describe the following set in Set-Builder form

{0, ±1, ±2, ±3}

Describe the following set in Set-Builder form

`{1/2, 2/5, 3/10, 4/17, 5/26, 6/37, 7/50}`

Describe the following set in Set-Builder form

{0, –1, 2, –3, 4, –5, 6, ...}