Test: Sequences & Summations - Computer Science Engineering (CSE) MCQ

Expand the sequence ⌈n/2⌉+⌊n/2⌋ where a1 is ⌊0.5⌋+⌈0.5⌉ = 1+0 = 1, a2 is ⌊1⌋+⌈1⌉ = 1 + 1 = 2 and so on.
⌈n/2⌉ represent greatest integer function
⌊n/2⌋  represent ceiling function

If there is one-to-one correspondence then they have same cardinality.

The value of ∑(i=1)³ ∑(h=0)² i = 1+1+1+2+2+2+3+3+3 = 18.

First five term of the sequence n! is given by 1, 1, 2, 6, 24.

The product of a₁, a₂, a₃ …… a_n is represented by ∏(i=1)ⁿ a_i.

The ratio of consecutive numbers is close to 3. Comparing these terms with the sequence of {3ⁿ} which is 3, 9, 27 …. Comparing these terms with the corresponding terms of sequence {3ⁿ} and the nth term is 2 less than the corresponding power of 3.

Using the formula. ∑(k=1)ⁿ k² = (n(n + 1)(2n + 1)) / 6.

a₇ = ⌊√14+1/2⌋ which is ⌊4.24⌋ = 4.

Put n = 4 in the sequence.

There is one-to-one correspondence between set of positive integers and set of all integers.