MATHEMATICAL INDUCTION.

 MATHEMATICAL  INDUCTION

A) Prove by the principal of induction:

1) 1²+ 2²+3²+....n²=n/16 (n+1((2n+1).

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

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

4) 1/1.2.3 + 1/2.3.4 +...+ 1/ n(n+1)(n+2) =n(n+3)/{4(n+1)(n+2)}.

5) (1-1/2)(1-1/3)(1-1/4)...(1- 1/(n+1) = 1/(n+1)

6) a+ar+ ar²+ ...arⁿ⁻¹= a(rⁿ -1)/(r-1)

7) x+ 4x +7x +..(3n-2)x=nx(3n-1)/2

8) 2+4+6+8+...2n = n(n+1)

9) 1+3+3²+ ...+3ⁿ⁻¹=1/2(3ⁿ-1)

10) 2+6+18+...2.3ⁿ⁻¹=(3ⁿ-1)

11) 1.2+2.2²+3.2³+... +n2ⁿ= (n -1) 2ⁿ⁺¹ +2

12) 3.2²+3²2³+3³2⁴+...+3ⁿ.2ⁿ⁺¹=12/5 (6ⁿ -1)

13) 1.3+3.5+5.7+..+(2n-1)(2n+1)= n(4n²+6n -1)/3

14) 5+15+45+....5(3)ⁿ⁻¹=5/2(3ⁿ-1)

15)1.1!+2.2!+3.3!+...+n.n!= (n+1)! - 1

 16) 1/(1.4) + 1/(4.7)+ 1/(7.10)+.... +1/{(3n-2)(3n+1) = n/(3n+1)

17) 1+5+12+22+35+......to n terms =  n²(n+1)/2

18)(1+3)(1+ 5/4)(1+7/9)...{1+ (2n+1)/n²} = (n+1)²

19) 

B) Prove by the principal of induction

1) n(n+1)(2n+1) is divisible by 6

2) n(n+1)(n+5) is multiple of 3

3) n(n²+20) is divisible by 48 if n is even.

4) 2³ⁿ-1 is divisible by 7.

5) 7ⁿ- 3ⁿ is divisible by 4 for all n belongs to N 

6) (10²ⁿ⁻¹+1) is divisible by 11

7) (2.7ⁿ+ 3. 5ⁿ -5) is divisible by 24.

8)3²ⁿ⁺²- 8n -9 is divisible by 8.

9)10ⁿ+3+3.4ⁿ⁺²+5 is divisible by 9

10) 3⁴ⁿ⁺²- 5²ⁿ⁺¹ is multiple of 14.

11) 7²ⁿ + (2³ⁿ⁻³)3ⁿ⁻¹ divisible by 25

12) 11ⁿ⁺²+ 12²ⁿ⁺¹ divisible by 133

13) (x²ⁿ-1) is divisible by (x-1) where x≠1.

14) {(41)ⁿ-(14)ⁿ} is a multiple of 27.

C) Prove by the principal of induction

17) 1+2+3....+n < (2n+1)²

18) 2ⁿ > n 

) 3ⁿ ≥ 2ⁿ 

) (1+2+3+..n)< 1/8 (2n+1)².