**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.

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

*a + ar +* *ar ^{2}+ …+ar^{n-1} = {a(r^{n}– 1)} / {(r – 1)}*

Solution:

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

P(n):* a + ar +* *ar ^{2}+ …+ar^{n-1} = {a(r^{n}– 1)} / {(r – 1)}*

For n = 1, we have ,

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

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

*a + ar +* *ar ^{2}+ …+ar^{n-1} = {a(r^{n}– 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.

**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)}/_{n}^{2}) = (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)}/_{n}^{2}) = (n+1)^{2}

For n = 1, we have

P(1): (1 + ^{3}/_{1}) = 4 = (1+1)^{2} = 2^{2} = 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)}/_{k}^{2}) = (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.

**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.

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

**1 ^{2} + 3^{2} + 5^{2} + … +(2n – 1)^{2} = ^{n(2n-1)(2n+1)}/_{3}**

Solution:

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

P(n): 1^{2} + 3^{2} + 5^{2} + … +(2n – 1)^{2} = ^{n(2n-1)(2n+1)}/_{3}

For n = 1, we have

P(1) = 1^{2} = 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) = 1^{2} + 3^{2} + 5^{2} + … +(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.

**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.

**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.

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