What is the sum of all odd numbers up to 240?

Given:

We need to find the sum of all odd numbers up to 240.

Concept Used:

The sum of the first n odd numbers is given by the formula S = n², where n is the number of terms.

Calculation:

We need to determine the number of odd numbers up to 240.

⇒ The sequence of odd numbers up to 240 is: 1, 3, 5, ..., 239

⇒ This is an arithmetic sequence with the first term a = 1 and the common difference d = 2.

⇒ The nth term of an arithmetic sequence is given by the formula: a_n = a + (n - 1)d

⇒ Setting the nth term to 239, we get: 239 = 1 + (n - 1) × 2

⇒ 239 = 1 + 2n - 2

⇒ 239 = 2n - 1

⇒ 240 = 2n

⇒ n = 120

So, there are 120 odd numbers up to 240.

⇒ Using the formula for the sum of the first n odd numbers:

⇒ S = n²

⇒ S = 120²

⇒ S = 14400

∴ The sum of all odd numbers up to 240 is 14400.