Breath Math

11: Prove the following by using the principle of mathematical induction for all n€ N:

¹/_1.2.3 + ¹/_2.3.4 + ¹/_3.4.5 + … +¹/_n(n+1)(n+2) = ⁿ⁽ⁿ⁺³⁾/_4(n+1)(n+2)

Solution:

Let the given statement be P(n), i.e.,

P(n): ¹/_1.2.3 + ¹/_2.3.4 + ¹/_3.4.5 + … +¹/_n(n+1)(n+2) = ⁿ⁽ⁿ⁺³⁾/_4(n+1)(n+2)

For n = 1, we have

P(1): ¹/_1.2.3 = ^1.(1+3)/_4(1+1)(1+2) = ^1.4/_4.2.3 = ¹/_1.2.3, which is true.

Let P(k) be true for some positive integer k, i.e.,

¹/_1.2.3 + ¹/_2.3.4 + ¹/_3.4.5 + … +¹/_k(k+1)(k+2) = ^k(k+3)/_4(k+1)(k+2)

We shall now prove that P(k + 1) is true.

Consider

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

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12. Prove the following by using the principle of mathematical induction for all n€ N:

a + ar + ar²+ …+ar^n-1 = {a(rⁿ– 1)} / {(r – 1)}

Solution:

Let the given statement be P(n), i.e.,

P(n): a + ar + ar²+ …+ar^n-1 = {a(rⁿ– 1)} / {(r – 1)}

For n = 1, we have ,

P(1): a = {a(r¹– 1)} / {(r – 1)}  = a which is true.

Let P(k) be true for some positive integer k, i.e.,

a + ar + ar²+ …+ar^n-1 = {a(rⁿ– 1)} / {(r – 1)}  ————(1)

We shall now prove that P(k + 1) is true.

Consider

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

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13: Prove the following by using the principle of mathematical induction for all n € N:

(1+ ³/₁) (1+ ⁵/₄) (1+ ⁷/₉)… (1+ ⁽²ⁿ⁺¹⁾/_n²) = (n+1)²

Solution:

Let the given statement be P(n), i.e.,

P(n): (1+ ³/₁) (1+ ⁵/₄) (1+ ⁷/₉)… (1+ ⁽²ⁿ⁺¹⁾/_n²) = (n+1)²

For n = 1, we have

P(1): (1 + ³/₁) = 4 = (1+1)² = 2² = 4, which is true

Let P(k) be true for some positive integer k, i.e.,

(1+ ³/₁) (1+ ⁵/₄) (1+ ⁷/₉)… (1+ ^(2k+1)/_k²) = (k+1)²

We shall now prove that P(k + 1) is true.

Consider

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

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14. Prove the following by using the principle of mathematical induction for all n € N:

(1 + ¹/₁)(1 + ¹/₂)(1 + ¹/₃)…(1 + ¹/_n) = (n + 1)

Solution:

Let the given statement be P(n), i.e.,

P(n): (1 + ¹/₁)(1 + ¹/₂)(1 + ¹/₃)…(1 + ¹/_n) = (n + 1)

For n = 1, we have ,

P(1): (1 + ¹/₁) = 2 = (1+1) which is true.

Let P(k) be true for some positive integer k, i.e.,

P(k): (1 + ¹/₁)(1 + ¹/₂)(1 + ¹/₃)…(1 + ¹/_k) = (k + 1) ————-(1)

We shall now prove that P(k + 1) is true.

Consider

Thus, P(k + 1) is true whenever P(k) is true. Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

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15: Prove the following by using the principle of mathematical induction for all n € N:

1² + 3² + 5² + … +(2n – 1)² = ^{n(2n-1)(2n+1)}/₃

Solution:

Let the given statement be P(n), i.e.,

P(n): 1² + 3² + 5² + … +(2n – 1)² = ^{n(2n-1)(2n+1)}/₃

For n = 1, we have

P(1) = 1² = 1 = ^{1(2.1-1)(2.1+1)}/₃ = ^1.1.3/₃ = 1, which is true.

Let P(k) be true for some positive ineteger k, i.e.,

P(k) = 1² + 3² + 5² + … +(2k – 1)² = ^{k(2k-1)(2k+1)}/₃ ———(1)

We shall now prove that P(k + 1) is true.

Consider

Thus, P(k + 1) is true whenever P(k) is true. Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

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16: Prove the following by using the principle of mathematical induction for all n€ N:

¹/_1.4 + ¹/_4.7 + ¹/_7.10 + … + ¹/_(3n-2)(3n+1) = ⁿ/_(3n+1)

Solution:

Let the given statement be P(n), i.e.,

P(n): ¹/_1.4 + ¹/_4.7 + ¹/_7.10 + … + ¹/_(3n-2)(3n+1) = ⁿ/_(3n+1)

For n = 1, we have

P(1): ¹/_1.4 = ¹/_3.1+1 = ¹/₄ = ¹/_1.4, which is true.

Let P(k) be true for some positive integer k, i.e.,

P(k): ¹/_1.4 + ¹/_4.7 + ¹/_7.10 + … + ¹/_(3k-2)(3k+1) = ^k/_(3k+1) ———-(1)

We shall now prove that P(k+1) is true.

Consider

Thus, P(k + 1) is true whenever P(k) is true. Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

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17. Prove the following by using the principle of mathematical induction for all n € N:

¹/_3.5 + ¹/_5.7 + ¹/_7.9 + … + ¹/_(2n+1)(2n+3) = ⁿ/_3(2n+3)

Solution:

Let the given statement be P(n), i.e.,

¹/_3.5 + ¹/_5.7 + ¹/_7.9 + … + ¹/_(2n+1)(2n+3) = ⁿ/_3(2n+3)

For n = 1, we have ,

P(1): ¹/_3.5 = ¹/_3(2.1+3) = ¹/_3.5 which is true.

Let P(k) be true for some positive integer k, i.e.,

¹/_3.5 + ¹/_5.7 + ¹/_7.9 + … + ¹/_(2k+1)(2k+3) = ^k/_3(2k+3) ————(1)

We shall now prove that P(k + 1) is true.

Consider

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

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