To find the sum $x$ of all integers from 51 to 100 inclusive, we can use the formula for the sum of an arithmetic series:

$S = \frac{n}{2} \times (a + l)$

Where:

• $n$ is the number of terms,
• $a$ is the first term, and
• $l$ is the last term.

Step 1: Determine the values of $a$, $l$, and $n$.

• The first term $a = 51$,

• The last term $l = 100$,

• To find $n$, the number of terms, use the formula for the $n$-th term of an arithmetic sequence:

  $n = l - a + 1 = 100 - 51 + 1 = 50.$

Step 2: Apply the formula for the sum of the arithmetic series.

Now that we know $n = 50$, $a = 51$, and $l = 100$, we can substitute these values into the sum formula:

$S = \frac{50}{2} \times (51 + 100) = 25 \times 151 = 3,775.$

Thus, the value of $x$ is 3,775.

Final Answer:

$x = 3,775.$

Would you like to see more about how this formula works or ask any questions? Here are some related questions:

1. What is the sum of integers from 1 to 100?
2. How can we calculate the sum of the squares of integers from 51 to 100?
3. How would the sum change if the range was from 51 to 99 instead?
4. What is the sum of all odd numbers between 51 and 100?
5. How can you use the sum formula for an arithmetic series to find the average of the numbers?

Tip: The sum of consecutive integers is a great example of an arithmetic sequence where using the formula can save a lot of time.