Final answer:
The sum of the first n odd natural numbers is found using telescoping sums to be 2n², which simplifies to a closed form solution of n². Therefore, the correct answer is (a) n².
Step-by-step explanation:
To find the closed form solution for the sum of the first n odd natural numbers using telescoping sums, we must first write the sum of n odd numbers in a general form which is 1 + 3 + 5 + ... + (2n-1). This sum can be manipulated by taking (n-1) from the last term and adding it to the first term, which gives us 2[n + (n-3) + (n-5) + ... + 3 + 1]. When we continue this process until all the terms in the sum are 'n', we will have n terms that are equal to n. This results in 2n2
A closed-form solution means we are looking for a simple expression that can represent the sum without actually having to sum all the individual terms. Applying the process above, we find the sum is 2n2, which after simplification gives us n2 as the closed form solution. Therefore, the correct answer for the sum of the first n odd natural numbers using telescoping sums is (a) n2.