Discrete Math. Practice. For students of technical specialties

Solution: we prove the inequality using the mathematical induction method: Induction

base: for n = 1 we have: 2> 1.

Induction step: let the inequality be valid for k = n, we prove the inequality for k = n +1: 2

> n+1

2*2

> n+1 (1) We

use the property: if a> b and b ≥ c, then a> c, i.e. we replace in (1) 2

with n, we have: 2n ≥ n+1

n ≥ 1 (2)

Studying inequality (2), we can conclude that 2

> n only for n ≥ 1, and since. n ∈ N, then this condition is satisfied, the inequality is proved.

Problem number 10: prove the equality: 1

+2

+3

+…+n

= ((1) \ (6)) n (n+1) (2n+1).

Solution: we prove the inequality using the mathematical induction method: Induction

base: for n = 1 we have: 1= ((1) \ (6)) *2*3=1

Induction step: let the inequality be valid for k = n, we prove the inequality for k = n +1: 1

+2

+3

+…+n

+ (n+1)

= ((1) \ (6)) (n+1) (n+1+1) (2 (n+1) +1)

1

+2

+3

+…+n

+ (n+1)

= ((1) \ (6)) (n+1) (n+2) (2n+3) (1)

Replace in (1) {1

+2

+3

+…+n

} by {((1) \ (6)) n (n+1) (2n+1)}, we get: ((1) \ (6)) n (n+1) (2n+1) + (n+1)

= ((1) \ (6)) (n+1) (n+2) (2n+3)

n (2n+1) +6 (n+1) = (n+2) (2n+3)

2n

+n+6n+6 = 2n

+3n+4n+6

2n

+7n+6 = 2n

+7n+6

Equality is proved.

Problem number 11: there is a chessboard, a horse is located in the lower left corner. Cut the bottom right corner from the board. Question: is it possible to get around such a chessboard with a knight?

Solution: it is easy to notice that, making a move with the knight, we alternate the color of the cell on which it is located, i.e. if we started with a black cell our movement, then we must finish on white. It is also clear that the left and right lower cells are not the same color. So at the last step we have 2 cells left (a horse is standing on one of them), which we have not yet walked around, and they are of the same color.

Therefore, it is impossible to get around such a chessboard with a horse.

Problem number 12: to prove the inequality: ((1) \ (√ 1)) + ((1) \ (√ 2)) + ((1) \ (√ 3)) + ((1) \ (√ 4)) +…+ ((1) \ (√ n)) + ≥ √ n for n≥0.

Solution: we prove the inequality using the method of mathematical induction:

Base of induction: for n = 1 we have:1=1.