Question

Find the largest integer not greater than the following expression :


`\displaystyle \large \boldsymbol{} (2150)/(2005) +(2150)/(2006) +(2150)/(2007 ) + ...+(2150)/(2020)`

Answer 1

• 17

Explanation:

The number of terms:

• 2020 - 2004 = 16

Each term is greater than 1, so the expression is greater than 16:

• 2150/2005 = 1 + 145/2005
• 2150/2006 = 1 + 144/2006
• ...
• 2150/2020 = 1 + 130/2020

It's easy to note each of the numbers getting smaller like 145/2005 > 144/2006 etc. Taking the smallest fraction.

The sum is:

• S > 16 + 16*130/2020 = 16 + 2080/2020 > 16 + 1 = 17

So the largest integer not greater than S is 17.