Answer :
To solve the problem where you need to find the value of [tex]\( x \)[/tex] in the sum of 89 consecutive integers from [tex]\(-43\)[/tex] to [tex]\( x \)[/tex], inclusive, given that the sum is 89, we can follow these steps:
1. Understand the Given Information:
- We have 89 consecutive integers starting from [tex]\(-43\)[/tex].
- The sum of these integers is 89.
2. Set Up the Arithmetic Sequence:
- An arithmetic sequence is defined by a starting number and a common difference. Here, the common difference is 1 (since the numbers are consecutive).
- Let [tex]\( x \)[/tex] be the last number in the sequence.
3. Find the Sum of the Arithmetic Sequence:
- The sum [tex]\( S \)[/tex] of an arithmetic sequence can be found using the formula:
[tex]\[
S = \frac{n}{2} \times (\text{first term} + \text{last term})
\][/tex]
where [tex]\( n \)[/tex] is the number of terms.
- Here, [tex]\( n = 89 \)[/tex], the first term is [tex]\(-43\)[/tex], and the last term is [tex]\( x \)[/tex].
4. Set Up the Equation:
- Plugging into the formula, we get:
[tex]\[
89 = \frac{89}{2} \times (-43 + x)
\][/tex]
5. Solve the Equation:
- Simplify the equation:
[tex]\[
89 = \frac{89}{2} \times (-43 + x)
\][/tex]
- Multiply both sides by 2 to eliminate the fraction:
[tex]\[
178 = 89 \times (-43 + x)
\][/tex]
- Divide both sides by 89:
[tex]\[
2 = -43 + x
\][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 2 + 43
\][/tex]
[tex]\[
x = 45
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\(\boxed{45}\)[/tex].
1. Understand the Given Information:
- We have 89 consecutive integers starting from [tex]\(-43\)[/tex].
- The sum of these integers is 89.
2. Set Up the Arithmetic Sequence:
- An arithmetic sequence is defined by a starting number and a common difference. Here, the common difference is 1 (since the numbers are consecutive).
- Let [tex]\( x \)[/tex] be the last number in the sequence.
3. Find the Sum of the Arithmetic Sequence:
- The sum [tex]\( S \)[/tex] of an arithmetic sequence can be found using the formula:
[tex]\[
S = \frac{n}{2} \times (\text{first term} + \text{last term})
\][/tex]
where [tex]\( n \)[/tex] is the number of terms.
- Here, [tex]\( n = 89 \)[/tex], the first term is [tex]\(-43\)[/tex], and the last term is [tex]\( x \)[/tex].
4. Set Up the Equation:
- Plugging into the formula, we get:
[tex]\[
89 = \frac{89}{2} \times (-43 + x)
\][/tex]
5. Solve the Equation:
- Simplify the equation:
[tex]\[
89 = \frac{89}{2} \times (-43 + x)
\][/tex]
- Multiply both sides by 2 to eliminate the fraction:
[tex]\[
178 = 89 \times (-43 + x)
\][/tex]
- Divide both sides by 89:
[tex]\[
2 = -43 + x
\][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 2 + 43
\][/tex]
[tex]\[
x = 45
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\(\boxed{45}\)[/tex].
