Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) [English] 11 Standard Maharashtra State Board chapter 4 - Methods of Induction and Binomial Theorem [Latest edition]

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

2 + 4 + 6 + ..... + 2n = n (n+1)

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

3 + 7 + 11 + ..... + to n terms = n(2n+1)

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

1² + 2² + 3² + .... + n² = `("n"("n" + 1)(2"n" + 1))/6`

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

1² + 3² + 5² + .... + (2n − 1)² = `"n"/3 (2"n" − 1)(2"n" + 1)`

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

1³ + 3³ + 5³ + .... to n terms = n²(2n² − 1)

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

1.2 + 2.3 + 3.4 + ..... + n(n + 1) = `"n"/3 ("n" + 1)("n" + 2)`

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

1.3 + 3.5 + 5.7 + ..... to n terms = `"n"/3(4"n"^2 + 6"n" - 1)`

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

`1/(1.3) + 1/(3.5) + 1/(5.7) + ... + 1/((2"n" - 1)(2"n" + 1)) = "n"/(2"n" + 1)`

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

`1/(3.5) + 1/(5.7) + 1/(7.9) + ...` to n terms = `"n"/(3(2"n" + 3))`

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

(2³ⁿ − 1) is divisible by 7

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

(2⁴ⁿ−1) is divisible by 15

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

3ⁿ − 2n − 1 is divisible by 4

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

Expand: `(sqrt(3) + sqrt(2))^4`

Expand: `(sqrt(5) - sqrt(2))^5`

Expand: (2x² + 3)⁴ 

Expand: `(2x - 1/x)^6`

Find the value of `(sqrt(3) + 1)^4- (sqrt(3) - 1)^4`.

Find the value of `(2 + sqrt(5))^5 + (2 - sqrt(5))^5`

Prove that `(sqrt(3) + sqrt(2))^6 + (sqrt(3) - sqrt(2))^6` = 970

Prove that `(sqrt(5) + 1)^5 - (sqrt(5) - 1)^5` = 352

Using binomial theorem, find the value of (102)⁴ 

Using binomial theorem, find the value of (1.1)⁵ 

Using binomial theorem, find the value of (9.9)³ 

Using binomial theorem, find the value of (0.9)⁴ 

Without expanding, find the value of (x + 1)⁴ − 4(x + 1)³ (x − 1) + 6 (x + 1)² (x − 1)² − 4(x + 1) (x − 1)³ + (x − 1)⁴

Without expanding, find the value of (2x − 1)⁴ + 4(2x − 1)³ (3 − 2x) + 6(2x − 1)² (3 − 2x)² + 4(2x − 1)¹ (3 − 2x)³ + (3 − 2x)⁴ 

Find the value of (1.02)⁶, correct upto four places of decimal

Find the value of (1.01)⁵, correct up to three places of decimals.

Find the value of (0.9)⁶, correct upto four places of decimal

In the following expansion, find the indicated term.

`(2x^2 + 3/(2x))^8`, 3^rd term

In the following expansion, find the indicated term.

`(x^2 - 4/(x^3))^11`, 5^th term

In the following expansion, find the indicated term.

`((4x)/5 - 5/(2x))^9`, 7^th term

In the following expansion, find the indicated term.

`(1/3 + "a"^2)^12`, 9^th term

In the following expansion, find the indicated term.

`(3"a" + 4/"a")^13`, 10^th term

In the following expansion, find the indicated coefficient.

x³ in `(x^2 + (3sqrt(2))/x)^9`

In the following expansion, find the indicated coefficient.

x⁸ in `(2x^5 - 5/x^3)^8`

In the following expansion, find the indicated coefficient.

x⁹ in `(1/x + x^2)^18`

In the following expansion, find the indicated coefficient.

x^–3 in `(x - 1/(2x))^5`

In the following expansion, find the indicated coefficient.

x^–20 in `(x^3 - 1/(2x^2))^15`

Find the constant term (term independent of x) in the expansion of `(2x + 1/(3x^2))^9`

Find the constant term (term independent of x) in the expansion of `(x - 2/x^2)^15`

Find the constant term (term independent of x) in the expansion of `(sqrt(x) - 3/x^2)^10`

Find the constant term (term independent of x) in the expansion of `(x^2 - 1/x)^9`

Find the constant term (term independent of x) in the expansion of `(2x^2 - 5/x)^9`

Find the middle term in the expansion of `(x/y + y/x)^12`

Find the middle terms in the expansion of `(x^2 + 1/x)^7`

Find the middle term in the expansion of `(x^2 - 2/x)^8`

Find the middle term in the expansion of `(x/"a" - "a"/x)^10`

Find the middle terms in the expansion of `(x^4 - 1/x^3)^11`

In the expansion of (k + x)⁸, the coefficient of x⁵ is 10 times the coefficient of x⁶. Find the value of k.

Find the term containing x⁶ in the expansion of (2 − x) (3x + 1)⁹

The coefficient of x² in the expansion of (1 + 2x)^m is 112. Find m

State, by writing first four terms, the expansion of the following, where |x| < 1

(1 + x)⁻⁴

State, by writing first four terms, the expansion of the following, where |x| < 1

`(1 - x)^(1/3)`

State, by writing first four terms, the expansion of the following, where |x| < 1

(1 – x²)^–3

State, by writing first four terms, the expansion of the following, where |x| < 1

`(1 + x)^(-1/5)`

State, by writing first four terms, the expansion of the following, where |x| < 1

(1 + x²)^–1

State, by writing first four terms, the expansion of the following, where |b| < |a|

(a − b)⁻³ 

State, by writing first four terms, the expansion of the following, where |b| < |a| 

(a + b)⁻⁴ 

State, by writing first four terms, the expansion of the following, where |b| < |a| 

`("a" + "b")^(1/4)`

State, by writing first four terms, the expansion of the following, where |b| < |a| 

`("a" - "b")^(-1/4)`

State, by writing first four terms, the expansion of the following, where |b| < |a| 

`("a" + "b")^(-1/3)`

Simplify first three terms in the expansion of the following

(1 + 2x)^–4 

Simplify first three terms in the expansion of the following

`(1 + 3x)^(-1/2)`

Simplify first three terms in the expansion of the following

`(2 - 3x)^(1/3)`

Simplify first three terms in the expansion of the following

`(5 + 4x)^(-1/2)`

Simplify first three terms in the expansion of the following

`(5 - 3x)^(-1/3)`

Use binomial theorem to evaluate the following upto four places of decimal

`sqrt(99)`

Use binomial theorem to evaluate the following upto four places of decimal

`root(3)(126)`

Use binomial theorem to evaluate the following upto four places of decimal

`root(4)(16.08)`

Use binomial theorem to evaluate the following upto four places of decimal

(1.02)^–5 

Use binomial theorem to evaluate the following upto four places of decimal

(0.98)^–3 

Show That C₀ + C₁ + C₂ + .... C₈ = 256

Show That C₀ + C₁ + C₂ + .... C₉ = 512

Show That C₁ + C₂ + C₃ + .... C₇ = 127

Show That C₁ + C₂ + C₃ + .... C₆ = 63

Show That C₀ + C₂ + C₄ + C₆ + C₈ = C₁ + C₃ + C₅ + C₇ = 128

Show That C₁ + C₂ + C₃ + .... C_n = 2ⁿ − 1

Show That C₀ + 2C₁ + 3C₂ + 4C₃ + ... + (n + 1)C_n = (n + 2)2ⁿ⁻¹

Select the correct answer from the given alternatives.

The total number of terms in the expression of (x + y)¹⁰⁰ + (x − y)¹⁰⁰ after simplification is:

Select the correct answer from the given alternatives.

The middle term in the expansion of (1 + x)²ⁿ will be :

Select the correct answer from the given alternatives.

In the expansion of (x² − 2x)¹⁰, the coefficient of x¹⁶ is

Select the correct answer from the given alternatives.

The term not containing x in expansion of `(1 - x)^2 (x + 1/x)^10` is 

Select the correct answer from the given alternatives.

The number of terms in expansion of (4y + x)⁸ − (4y − x)⁸ 

Select the correct answer from the given alternatives.

The value ¹⁴C₁ + ¹⁴C₃ + ¹⁴C₅ + ..... + ¹⁴C₁₁ is

Select the correct answer from the given alternatives.

The value ¹¹C₂ + ¹¹C₄ + ¹¹C₆ + ¹¹C₈ is equal to

Select the correct answer from the given alternatives.

In the expansion of (3x + 2)⁴, the coefficient of the middle term is

Select the correct answer from the given alternatives.

The coefficient of the 8^th term in the expansion of (1 + x)¹⁰ is:

Select the correct answer from the given alternatives.

If the coefficient of x² and x³ in the expansion of (3 + ax)⁹ are the same, then the value of a is

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"`

Expand (3x² + 2y)⁵ 

Answer the following:

Expand `((2x)/3 - 3/(2x))^4`

Answer the following:

Find third term in the expansion of `(9x^2 - y^3/6)^4`

Answer the following:

Find tenth term in the expansion of `(2x^2 + 1/x)^12`

Answer the following:

Find the middle term (s) in the expansion of `((2"a")/3 - 3/(2"a"))^6`

Answer the following:

Find the middle term (s) in the expansion of `(x - 1/(2y))^10`

Answer the following:

Find the middle term (s) in the expansion of (x²+ 2y²)⁷ 

Answer the following:

Find the middle term (s) in the expansion of `((3x^2)/2 - 1/(3x))^9`

Find the coefficients of x⁶ in the expansion of `(3x^2 - 1/(3x))^9`.

Find the coefficients of x⁶⁰ in the expansion of `(1/x^2 + x^4)^18`

Answer the following:

Find the constant term in the expansion of `((4x^2)/3 + 3/(2x))^9`

Answer the following

Find the constant term in the expansion of `(2x^2 - 1/x)^12`

Answer the following:

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

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

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

Answer the following:

If the coefficient of x¹⁶ in the expansion of (x² + ax)¹⁰ is 3360, find a

Answer the following:

If the middle term in the expansion of `(x + "b"/x)^6` is 160, find b

Answer the following:

If the coefficient of x² and x³ in the expansion of (3 + kx)⁹ are equal, find k

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If the constant term in the expansion of `(x^3 + "k"/x^8)^11` is 1320, find k

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Show that there is no term containing x⁶ in the expansion of `(x^2 - 3/x)^11`

Answer the following:

Show that there is no constant term in the expansion of `(2x - x^2/4)^9`

Answer the following:

State, first four terms in the expansion of `(1 - (2x)/3)^(-1/2)`

Answer the following:

State, first four terms in the expansion of `(1 - x)^(-1/4)`

Answer the following:

State, first three terms in the expansion of `(5 + 4x) ^(-1/2)`

Answer the following:

Using binomial theorem, find the value of `root(3)(995)` upto four places of decimals

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Find approximate value of `1/4.08` upto four places of decimals

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Find the term independent of x in the in expansion of `(1 - x^2) (x + 2/x)^6`

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(a + bx) (1 − x)⁶ = 3 − 20x + cx² + ..... then find a, b, c

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The 3^rd term of (1 + x)ⁿ is 36x². Find 5^th term

Answer the following:

Suppose (1 + kx)ⁿ = 1 − 12x + 60x² − .... find k and n.