Prove the following by using the principle of mathematical induction for all n∊ N 1+3+3²+.....+3^n-1 = \(\frac { ({ 3 }^{ n }-1) }{ 2 } \)

5

(a)

Prove the following by using the principle of mathematical induction for all n ∊ N:1.2.3+2.3.4+...+n(n+1)(n+2)= \(\frac { n(n+1)(n+2)(n+3) }{ 4 } \)

5

(a)

Prove the following by using the principle of mathematical induction for all n ∊ N: 1.3+3.5+5.7+....+(2n-1)(2n+1) = \(\frac { n(4{ n }^{ 2 }+6n-1) }{ 3 } \)

5

(a)

Prove the following by using the principle of mathematical induction for all n ∊ N \(\frac { 1 }{ 2 } +\frac { 1 }{ 4 } +\frac { 1 }{ 8 } +....+\frac { 1 }{ 2^{ n } } =1-\frac { 1 }{ 2^{ n } } \)

5

(a)

Prove the following by using the principle of mathematical induction for all n ∊ N:a+ar+ar2+.......+ar^n-1 =\(\frac { a({ r }^{ n }-1) }{ r-1 } \)

5

(a)

Prove the following by using the principle of mathematical induction for all n ∊ N: 1²+3²+5²+....+(2n-1)² = \(\frac { n(2n-1)(2n+1) }{ 3 } \)

5

(a)

Prove the following by using the principle of mathematical induction for all n ∊ N:1+2+3+... +\(n<\frac { 1 }{ 8 } (2n+1)^{ 2 }\)

5

(a)

Prove the following by using the principle of mathematical induction for all n ∊ N: x²ⁿ-y²ⁿ is divisible by (x+y)

5

(a)

Prove the following by using the principle of mathematical induction for all n∊ N (2n+7) < (n+3)²

5

(a)

Prove by the principle of mathematical induction that for all n∊N \(\frac { 1 }{ 1.2 } +\frac { 1 }{ 2.3 } +\frac { 1 }{ 3.4 } +...+\frac { 1 }{ n(n+1) } =\frac { n }{ (n+1) } \)

5

(a)