**Prove the following through principle of mathematical induction for all values of n, where n is a natural number.**

**1) 1+3+32+….+3n–1=(3n–1)2**

** **

**Sol:**

**The given statement is:**

**P(n) :**

**Now, for n = 1**

P(1) = **= 1**

**Thus, the P(n) is true for n=1**

**Let, [2(k+1)+7]=[(2k+7)+2][2(k+1)+7]=[(2k+7)+2][2(k+1)+7]=[(2k+7)+2]**

P(k) be true, where k is a positive integer.

**. . . . . . . . . . (1)**

**Now, we will prove that P(k+1) is also true:**

**P(k + 1):**

**=**

=**[Using equation (1)]**

=

=

=

=

=

**Thus, whenever** **P(k) proves to be true**, **P(k+1) subsequently proves to be true**.

**Therefore, with the help of mathematical induction principle it can be proved that the statement P(n) is true for all the natural numbers.**

**2: 13+23+33+……+n3 = (n(n+1)2)2**

** **

**Sol:**

**The given statement is:**

**P(n):**

**Now, for n = 1**

**P(1):** **= 1**

**Thus, the P(n) is true for n = 1**

**Let, P(k) be true, where ‘k’ is a positive integer**.

**. . . . . . . . . . . . . . . . . (1)**

**Now, we will prove P(k + 1) is also true.**

**P(k + 1):**

=

=

=** [From equation (1)]**

=

=

=

=

=

=

=

**Thus, whenever P(k) proves to be true, P(k + 1) subsequently proves to be true.**

**Therefore, with the help of mathematical induction principle it can be proved that the statement P(n) is true for all the natural numbers.**

**3: 1+11+2+11+2+3+….+11+2+3+…+n=2nn+1**

** **

**Sol:**

**The given statement is:**

**P(n):**

**Now, for n = 1**

**P(1):**

**Thus, the P(n) is true for n=1**

**Let, P(k) be true, where k is a positive integer.**

**. . . . . . . . . . . . . . (1) **

**Now, we will prove P(k + 1) is also true.**

=

=

=**[From equation (1)]
**

= **( 1+2+3+…+n=n(n+1)2)**

=

=

=

=

=

**Thus, whenever P(k) proves to be true, P(k + 1) subsequently proves to be true.**

**Therefore, with the help of mathematical induction principle it can be proved that the statement P(n) is true for all the natural numbers.**

**4: 1.2.3+2.3.4+…+n(n+1)(n+2)=n(n+1)(n+2)(n+3)4**

**Sol:**

**The given statement is:**

**P(n):**

**Now, for n = 1**

**P(1):**

**Thus, the P(n) is true for n=1.**

**Let, P(k) be true, where k is a positive integer.**

**. . . . . . . . . . (1)**

**Now, we will prove P(k + 1) is true**.

=

=**[By using equation (1)]**

=

**Now, by using equation (1) :**

=

=

=

**Thus, whenever P(k) proves to be true, P(k + 1) subsequently proves to be true.**

**5: 1.3+2.32+3.33+…+n.3n=(2n−1)3n+1+34**

** **

**Sol:**

**The given statement is:**

**P(n):**

**Now, for n = 1:**

=

**Thus, the P(n) is true for n=1.**

**Let, P(k) be true, where k is a positive integer.**

1.3 + 2. 3^{2}+ 3.3^{3} +…+ k. 3^{k} = \frac{(2k-1)3^{k+1}\, +\, 3}{4} **. . . . . . . (1)**

**Now, we will prove P(k + 1) is also true**:

=

=**[By using equation (1)]**

=

=

=

=

=

=

**Thus, whenever P(k) proves to be true, P(k + 1) subsequently proves to be true.**

**6: 1.2+2.3+3.4+…+n.(n+1)=[n(n+1)(n+2)3]**

** **

**Sol:**

**The given statement is:**

**P(n):**

**Now, for n = 1:**

=

**Thus, the P(n) is true for n=1.**

**Let, P(k) be true, where k is a positive integer.**

** . . . . . . . . . (1) **

**Now, we will prove P(k + 1) is also true:**

=

=

**Now, by using equation (1):**

=

=

=

**Thus, whenever P(k) proves to be true, P(k + 1) subsequently proves to be true.**

**7: 1.3+3.5+5.7+…+(2n–1)(2n+1)=n(4n2+6n–1)3**

**Sol:**

**The given statement is:**

**Now, for n = 1:**

**= 3**

**Thus, the P(n) is true for n = 1**

**Let, P(k) be true, where k is a positive integer.**

**. . . . . . . . . (1)**

**Now we will prove P(k + 1) is also true:**

=

=

**Now, by using equation (1):**

=

=

=

=

=

=

=

=

=

=

=

**Thus, whenever P(k) proves to be true, P(k + 1) subsequently proves to be true.**

**8: 1.2+2.22+3.22+…+n.2n=(n–1)2n+1+2**

**Sol:**

**The given statement is:**

**P(n):**

**Now, for n = 1:**

=

**Thus, the P(n) is true for n=1.**

**Let P(k) be true, where k is a positive integer:**

**. . . . . . . . (1)**

**Now we will prove P(k + 1) is also true:**

=

=

=

=

=

=

**Thus, whenever P(k) proves to be true, P(k + 1) subsequently proves to be true.**

** **

**9: 12+14+18+…+12n=1–12n**

**Sol:**

**The given statement is:**

**Now, for n = 1:**

=

**Thus, the P (n) is true for n = 1.**

**Let, P (k) be true, where k is a positive integer:**

** . . . . . . (1)**

**Now, we will prove P (k + 1) is also true:**

=

=

**Now, by using equation (1):**

=

=

=

=

**Thus, whenever P(k) proves to be true, P(k + 1) subsequently proves to be true**.

** 10: Provide a proof for the following with the help of mathematical induction principle for all values of n, where it is a natural number.**

** **

**Solution:**

**The given Statement is:**

**Q(n):**

**Now, for n = 1**

**Thus, Q(1) proves to be true.**

**Let’s assume Q(p) is true, where p is a natural number.**

**Q(p):**

** =** **. . . . . . . . . . . . . . (1)**

**Now, we have to prove that Q(p+1) is also true.**

**Since, Q (p) is true, we have:**

**Q (p+1):**

** =**

**Now, by using equation (1):**

=

=

=

=

=

=

=

=

=

=

**Thus, whenever** **Q (p) proves to be true, Q (p+1) subsequently proves to be true.**

**Therefore, with the help of induction principle it can be proved that the statement Q(n) is true for all the natural numbers.**

**11: Provide a proof for the following with the help of mathematical induction principle for all values of n, where it is a natural number.**

** **

**Solution:**

**The given statement is:**

**Q(n):**

**Now, for n = 1:**

**Thus, Q(1) proves to be true.**

**Let’s assume Q(p) is true, where p is a natural number:**

**Q (p):**

**=****. . . . . . . . . . . (1) **

**Now, we have to prove that Q (p+1) is also true.**

**Since, Q (p) is true, we have**:

**Q (p + 1):**

**=**

**Now, using equation (1):**

=

=

=

=

=

=

=

=

=

**Thus, whenever** **Q(p) proves to be true, Q(p+1) subsequently proves to be true.**

**Therefore, with the help of induction principle it can be proved that the statement Q(n) is true for all the natural numbers.**

**12: ****Provide a proof for the following with the help of mathematical induction principle for all values of n, where it is a natural number.**

** **

**Solution:**

**The given statement is:**

**Q(n):**

**Now, for n = 1**

**Thus, Q (1) proves to be true**.

**Let’s assume Q (p) is true, where p is a natural number.**

**Q (p) =** ** . . . . . . . . (1)**

**Now, we have to prove that Q (p+1) is also true.**

**Since, Q (p) is true, we have:**

Q(p+1) =

**Now, using Equation (1):**

=

=

=