Mathematical Foundation of Computer Science

DHARM

DISCRETE THEORY, RELATIONS AND FUNCTIONS 23

1.26 Prove that the sum of cubes of the first n natural numbers is equal to

Rnn()+

S

T

UV

W

1

2

2

.

1.27 Prove that for every n ≥ 0, 1 + ii n

i

n

*! (=+)!

=

∑^1

1

1.28 Prove that for every n ≥ 21 +

1

1 i

x

i

n

>

=

∑

1.29 Prove by mathematical Induction for every n ≥ 0,

2 + 2^2 + 2^3 + ... + 2n = 2(2n – 1)

1.30 Show that n^3 + 2n is divisible by 3 for every n ≥ 0.

1.31 Prove by Induction that for every n ≥ 1, the number of subsets of {1, 2, ..., n} is 2n.