B. Arithmetic Series
Completion requirements
B. Arithmetic Series
From Lesson 1.1, you know that an
arithmetic sequence
is a sequence whose consecutive
terms
have a common difference
. This definition is still valid when applied to an arithmetic series. The change is that a series refers to the terms in a sequence being added together. In other words, an arithmetic series is the sum of the
terms in an arithmetic sequence
.
As shown in Section A, Gauss found the formula for the sum of an arithmetic series to be
\[
S_n = \frac{n}{2}\left[ {2t_1 + (n - 1)d} \right]
\]
Take a closer look at this formula. Rewriting it, you can separate out \(t_1 \) and \(t_n \) to get a different variation of the formula.
\[
\begin{align}
S_n &= \frac{n}{2}\left[ {2t_1 + \left( {n - 1} \right)d} \right] \\
S_n &= \frac{n}{2}\left[ {t_1 + \left[ {t_1 + (n - 1)d} \right]} \right] \\
\end{align}
\]
Since \(t_n = t_1 + \left( {n - 1} \right)d \)
It follows that \(S_n = \frac{n}{2}\left( {t_1 + t_n } \right)\)
The formula you use will depend on the information that is provided in the problem.
As shown in Section A, Gauss found the formula for the sum of an arithmetic series to be
\[
S_n = \frac{n}{2}\left[ {2t_1 + (n - 1)d} \right]
\]
Take a closer look at this formula. Rewriting it, you can separate out \(t_1 \) and \(t_n \) to get a different variation of the formula.
\[
\begin{align}
S_n &= \frac{n}{2}\left[ {2t_1 + \left( {n - 1} \right)d} \right] \\
S_n &= \frac{n}{2}\left[ {t_1 + \left[ {t_1 + (n - 1)d} \right]} \right] \\
\end{align}
\]
Since \(t_n = t_1 + \left( {n - 1} \right)d \)
It follows that \(S_n = \frac{n}{2}\left( {t_1 + t_n } \right)\)
The formula you use will depend on the information that is provided in the problem.