Principle of mathematical induction Section- 1

1.For all n∈N, 3n5 + 5n³ + 7n is divisible by

1. None
2. 10
3. 5
4. 15

2. {1 – (1/2)}{1 – (1/3)}{1 – (1/4)} ……. {1 – 1/(n + 1)} =

1. 1/(n + 1) for all n ∈ R
2. 1/(n + 1) for all n ∈ N.
3. n/(n + 1) for all n ∈ N.
4. n/(n + 1) for all n ∈ R

3.For all n ∈ N, 32n + 7 is divisible by

1. 8
2. 4
3. 3
4. None of these

4. The sum of the series 1 + 2 + 3 + 4 + 5 + ………..n is

1. n/2
2.(n + 1)/2
3. n(n + 1)/2
4. None of these

5.The sum of the series 1² + 2² + 3² + ……….. n² is

1.n(n + 1) (2n + 1)/3
2. n(n + 1) (2n + 1)/6
3. n(n + 1) (2n + 1)/2
4. None of these

6.For all positive integers n, the number n(n² − 1) is divisible by:

1. 5
2. 3
3.6
4. 24

7.If n is an odd positive integer, then aⁿ + bⁿ is divisible by :

1. a – b
2. a+b
3. a² + b²
4. None of these.

8.n(n + 1) (n + 5) is a multiple of ____ for all n ∈ N

1. 3
2. 2
3. 6
4. None of these

9.For any natural number n, 7ⁿ – 2ⁿ is divisible by

1. 2
2. 4
3. 8
4. 5

10.The sum of the series 1³ + 2³ + 3³ + ………..n³ is

1. {n(n + 1)/2}²
2. n(n + 1)/2
3. {n/2}²
4. None of these

11.(1² + 2² + …… + n²) _____ for all values of n ∈ N

1. less than n³/3
2. None of these
3. = n³/3
4. > n³/3

12. {1/(3 ∙ 5)} + {1/(5 ∙ 7)} + {1/(7 ∙ 9)} + ……. + 1/{(2n + 1) (2n + 3)} =

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

13. If n is an odd positive integer, then aⁿ + bⁿ is divisible by :

1. a-b
2. a² + b²
3. a+b
4. None of these

14.(2 ∙ 7N + 3 ∙ 5N – 5) is divisible by ……….. for all N ∈ N

1. 21
2. 28
3. 25
4. 29

15. For all n∈N, 52n − 1 is divisible by

1. 24
2. 34
3. 36
4. None of the above