📂Lemmas

Proof that the Partial Sums of a Geometric Sequence are Also Geometric

Theorem

The geometric sequence $a_n = a r^{n-1}$, its partial sum $\displaystyle S_n = \sum_{k=1}^{n} a_k$, and some natural number $m$, such that $A_n = S_{mn} - S_{m(n-1)}$ is a geometric sequence.

Explanation

It’s really difficult if you don’t know.

For example, consider the sequence obtained by summing sets of three powers of 2, $(1 + 2+ 4)= 7$, $(8 + 16 + 32)=56$, $(64+128+256)=448 \cdots$ is a geometric sequence with the first term 7 and common ratio 8.

This property is also possessed by arithmetic sequences. The principle is actually simple, so read carefully once and then just remember the fact from the next time.

Proof

$$A_n = S_{mn} - S_{m(n-1)} = ar^{mn-1} + ar^{mn-2} + \cdots + ar^{mn-m}$$ By grouping the terms with respect to $a r^{mn-m}$ and rearranging the equation, $$A_n = a r^{mn-m} ( r^{m-1} + r^{m-2} + \cdots + 1) = a { {r^{m} - 1} \over {r-1} } \left( r^m \right) ^{n-1}$$ Therefore, $A_n$ is a geometric sequence with the first term $\displaystyle a { {r^{m} - 1} \over {r-1} }$ and the common ratio $r^{m}$. It’s not necessary to know exactly what the first term and common ratio are.

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