To find the common difference [tex]\( d \)[/tex] of an arithmetic sequence given the first term [tex]\( a_1 = 23 \)[/tex], the number of terms [tex]\( n = 15 \)[/tex], and the sum of the first 15 terms [tex]\( S_{15} = 765 \)[/tex], we can use the formula for the sum of the first [tex]\( n \)[/tex] terms of an arithmetic sequence:
[tex]\[
S_n = \frac{n}{2} \left(2a_1 + (n-1)d\right)
\][/tex]
In this case, we are given:
[tex]\[ S_{15} = 765 \][/tex]
[tex]\[ n = 15 \][/tex]
[tex]\[ a_1 = 23 \][/tex]
Substitute the given values into the formula:
[tex]\[
765 = \frac{15}{2} \left(2 \cdot 23 + (15-1)d\right)
\][/tex]
We'll simplify this step-by-step:
1. Evaluate the term inside the parentheses:
[tex]\[
765 = \frac{15}{2} \left(46 + 14d\right)
\][/tex]
2. Multiply both sides of the equation by 2 to eliminate the fraction:
[tex]\[
1530 = 15 \left(46 + 14d\right)
\][/tex]
3. Distribute the 15:
[tex]\[
1530 = 690 + 210d
\][/tex]
4. Subtract 690 from both sides to solve for [tex]\( 210d \)[/tex]:
[tex]\[
1530 - 690 = 210d
\][/tex]
[tex]\[
840 = 210d
\][/tex]
5. Divide both sides by 210 to find [tex]\( d \)[/tex]:
[tex]\[
d = \frac{840}{210}
\][/tex]
Thus, the common difference [tex]\( d \)[/tex] is:
[tex]\[
d = 4.0
\][/tex]
So, the common difference in the arithmetic sequence is [tex]\( d = 4.0 \)[/tex].
