Answer:
The sum of the first ten terms is -35.
Step-by-step explanation:
The first term of an arithmetic sequence is 10, and its common difference is -3.
We want to find the sum of the first 10 terms of this sequence.
Remember that the sum for an arithmetic sequence is given by:
[tex]\displaystyle S=\frac{k}{2}\left(a+x_k\right)[/tex]
Where k is the number of terms, a is the initial/first term, and x_k is the last term.
Since our initial term is 10, a = 10.
And since we want to find the sum of the first ten terms, k = 10.
So, we will need to find the last or tenth term. We can write an explicit formula. The standard explicit formula for an arithmetic sequence is:
[tex]x_n=a+d(n-1)[/tex]
Where a is the initial term and d is the common difference.
So, by substituting, we acquire:
[tex]x_n=10-3(n-1)[/tex]
Then the tenth or last term is:
[tex]x_{10}=10-3(10-1)=10-3(9)=10-27= -17[/tex]
Then the sum of the first ten terms will be:
[tex]\displaystyle S_{10}=\frac{10}{2}(10+(-17))=5(-7)=-35[/tex]
