Let us consider the geometric series \( \sum_{k=t}^m ar^{k} \). The first \( n \) partial sums are

\(S_t = a r^t\)
\(S_{ t + 1 } = a r^t + a r^{ t + 1 }\)
\(S_{ t + 2 } = a r^t + a r^{ t + 1 } + a r^{ t + 2 }\)
\( \vdots \)
\(S_n = a r^t + a r^{ t + 1 } + a r^{ t + 2 } + \cdots + a r^{ t + j }\)

A formula for \( S_n\) by using the following cleverness; Factor \( r^t \) from the nth partial sum yielding

$$S_n = r^t \left( a + a r^t + a r^{ t + 1 } + \cdots + a r^{ j } \right)$$

which is equal to

$$S_n = r^t \left( \sum_{k=0}^{ j } ar^{k} \right) = \frac{ r^t \left( a – a r^{ j + 1} \right)}{ 1 – r }$$

Now consider the geometric series \( \sum_{k=t}^m ar^{k – 1} \). The first \( n \) partial sums are

\(S_t = a r^{ t – 1 }\)
\(S_{ t + 1 } = a r^{ t – 1 } + a r^t \)
\(S_{ t + 2 } = a r^{ t – 1 } + a r^t + a r^{ t + 1 }\)
\( \vdots \)
\(S_n = a r^{ t – 1 } + a r^t + a r^{ t + 1 } + \cdots + a r^{ t + j – 1 }\)

A formula for \( S_n\) by using the following cleverness; Factor \( r^t \) from the \( n \)th partial sum yielding

$$S_n = r^t \left( a + a r^t + a r^{ t + 1 } + \cdots + a r^{ j – 1 } \right)$$

which is equal to

$$ S_n = r^t \left( \sum_{k=1}^{ j } ar^{k – 1} \right) = \frac{ r^t \left( a – a r^{ j } \right)}{ 1 – r } $$