Question

Give a Big-O estimate of the product of the first n odd numbers:

a) O(1)
b) O(logn)
c) O(n)
d) O(n2)

Answer 1

The Big-O estimate for the product of the first n odd numbers, which grows at most exponentially, is loosely bounded by
`O(n^2)` even though this is not the tightest bound.

Step-by-step explanation:

To give a Big-O estimate of the product of the first n odd numbers, first recognize that the nth odd number is given by the formula
`2n-1`. Therefore, the product of the first n odd numbers can be written as

`(1) × (3) × (5) × ... × (2n-3) × (2n-1)`. This product is known as the double factorial of n when n is odd, and its exact growth rate can be complex to analyze directly.

However, for a rough estimate, we can simply notice that each term in the product is at most 2n so the entire product is at most
`(2n)^n`. Since Big-O notation focuses on the upper bound of growth, we see that the growth of the product is at most exponential in n. Therefore, its Big-O estimate is not
`O(1), O(logn), or O(n)` but it is bounded by a function with an exponent, such as
`O(n^2)` which although is a loose upper bound, serves to represent exponential growth in this context.