Option A: [tex]-2500[/tex] is the value of [tex]S_{25[/tex]
Explanation:
It is given that the first term is [tex]a_1=20[/tex]
The common difference is [tex]d=-10[/tex]
We need to determine the sum of 25 terms.
The sum of terms of an arithmetic series can be determined using the formula, [tex]S_n=\frac{n[2a_1+(n-1)d]}{2}[/tex]
Substituting [tex]n=25[/tex] , [tex]a_1=20[/tex] and [tex]d=-10[/tex]
Thus, we have,
[tex]S_{25}=\frac{25[2(20)+(25-1)(-10)]}{2}[/tex]
Simplifying the values, we get,
[tex]S_{25}=\frac{25[40+(24)(-10)]}{2}[/tex]
[tex]S_{25}=\frac{25[40-240]}{2}[/tex]
Subtracting the terms within the bracket, we get,
[tex]S_{25}=\frac{25[-200]}{2}[/tex]
Multiplying the terms in the numerator, we have,
[tex]S_{25}=\frac{-5000}{2}[/tex]
Dividing, we get,
[tex]S_{25}=-2500[/tex]
Thus, the sum of the 25 terms is [tex]-2500[/tex]
Therefore, Option A is the correct answer.
