Question

The sum of x distinct integers greater than zero is less than 75. What is the greatest possible value of x?

Answer 1

Hint: Integers are the numbers that start from - infinity and go till +infinity.

Complete step-by-step answer:
Here, we are talking about only integers that are greater than zero means natural numbers only. Therefore, we can use the following formula that is
The formula for the sum of n natural numbers is
Sum \[=\dfrac{n\left( n+1 \right)}{2}\] .


As mentioned in the hint, the integers which are greater than zero means natural numbers only.

Hence, we can use the formula for getting the sum of natural numbers as it is given in the hint as follows

Sum \[=\dfrac{n\left( n+1 \right)}{2}\]

Now, we know that this sum should be less than 75, so, we can write as follows

\[\begin{align}
  & 75\ge \dfrac{x\left( x+1 \right)}{2} \\
 & x\left( x+1 \right)\le 150 \\
 & {{x}^{2}}+x-150\le 0 \\
\end{align}\]

Now, on solving the quadratic equation, we get

\[-13.25\le x\le 11.75\]

Now, we know that we have to see just the positive side as n is an integer which is greater than 0, so, we get

\[\begin{align}
  & x\le 11.75 \\
 & x=11 \\
\end{align}\]

Hence, the value of x is 11.

Note: The students can make an error by manually counting the positive integers or the integers that are greater than zero rather than using the summation formula for first n natural numbers which can be used and using it can make the problem very easy. Another thing that is so important to know the meaning of integers as well for correctly solving this question is that integers are the numbers that start from - infinity and go till +infinity.