Answer :
ANSWER
S₁₀ = 365
EXPLANATION
The sum of the first n terms of an arithmetic sequence is called the arithmetic series formula, given by,
[tex]S_n=\frac{n(a_1+a_n)}{2}[/tex]In this sequence, we can see that a₁ = 5, and the sum we have to find is the sum of the first 10 terms, so n = 10. To find the sum, we have to find the term a₁₀ first.
The nth term of an arithmetic sequence is given by the formula,
[tex]a_n=a_1+d(n-1)[/tex]Where d is the common difference. To find this formula for this sequence, we have to find the common difference by using any of the given terms. If we use n = 2 - in other words, we write it for a₂,
[tex]a_2=12=5+d(2-1)[/tex]Solving for d,
[tex]\begin{gathered} 12=5+d \\ d=12-5=7 \end{gathered}[/tex]Thus, the formula for the nth term of this sequence is,
[tex]a_n=5+7(n-1)[/tex]And the 10th term is,
[tex]a_{10}=5+7(10-1)=5+7\cdot9=5+63=68[/tex]So, the sum of the first 10 terms is,
[tex]S_{10}=\frac{10(5+68)}{2}=\frac{10\cdot73}{2}=\frac{730}{2}=365[/tex]Hence, the sum of the first 10 terms of this arithmetic sequence is 365.