<li><p>In an arithmetic sequence an, a7 = a3 + 18 = 4, which of the following equations is true?</p>
<p>(A) an = –5 + 4n
(B) an = –27.5 + 4n
(C) an = –5 + 4.5n
<a href=“D”>B</a> an = –27.5 + 4.5n<a href=“E”>/B</a> an = –27.5 + 6n</p></li>
</ol>
<p>Explanatio:n</p>
<p>In an arithmetic sequence, the difference between consecutive terms is constant. To find the formula for an arithmetic sequence, two values are needed: (1) the difference between consecutive terms, and (2) the value of a term. The given information easily tells both: the difference between consecutive terms follows from knowing the difference between the third term and the seventh term. If a7 = a3 + 18, then the difference between an+ 1 and an is 18/7 – 3 = 18/4 = 4.5. And the value of a term is given: a7 = 4. Thus, the formula for the sequence must satisfy the following equation: an = 4.5n + x. Since a7 = 4.5(7) + x = 4, it follows that x = 4 – 4.5(7) = –27.5. Therefore, the formula for the series is an = –27.5 + 4.5n.</p>
<p>I basically understand everything up until the bold, where they choose an equation. I know i can get the answer by simply plugging in, but how can i do it algebraically?
Is there any specific formula to find the equation of a sequence in this form? Why is it just n instead of (n-1)?</p>