There is a standard way to induce an arithmetic progression out of a given geometric progression, as

Question

There is a standard way to induce an arithmetic progression out of a given geometric progression, as illustrated from the following examination question.

The geometric sequence $u_1, u_2, u_3, \dots$ has common ratio $r$.

Consider the sequence $A = \{a_n = \log_2 |u_n| : n \in \mathbb{Z}^+\}$.

Show that $A$ is an arithmetic sequence, stating its common difference $d$ in terms of $r$.

A particular geometric sequence has $u_1 = 3$ and a sum to infinity of $4$.

Find the value of $d$.

Accepted Answer

(a) $d=\log_2|r|$ (b) $d=-2$

Connecting GP with AP and Summation

Connecting GP with AP and Summation

Solution:

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