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The sum of the first 15 terms of an arithmetic sequence is 765. If the first term is 23, what is the common difference [tex]d[/tex]?

Given:
[tex]a_1 = 23[/tex]
[tex]n = 15[/tex]
[tex]s_{15} = 765[/tex]


Sagot :

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].