Question

Can a sequence be both arithmetic and geometric? Explain why.

Answer 1

For an arithmetic sequence a_n, each successive term has a common difference d. For a geometric sequence b_n, each successive term has a common ratio r.

For a_n, the terms look like
\[\left\{a, a+d,a+2d,\ldots\right\}\]
and for A_n, you have
\[\left\{a,ar, ar^2,\ldots\right\}\]
Suppose a_n = A_n. That means that every term of a_n corresponds exactly with every term of A_n, meaning
\[a_1=A_1,\\
a_2=A_2,\\
\vdots\]
Under this assumption, you have
\[\begin{cases}a = a\\
a+d=ar\\
a+2d=ar^2\\
\vdots\end{cases}\]
I'm pretty sure there are no real numbers d and r that satisfy the system, so I don't think a sequence can be both arithmetic and geometric.

Answer 2

Except for d = 0 and r = 1, but then you just have the first term of both sequences, a.

Answer 3

But that is such a sequence.

Answer 4

I was hoping to arrive at some contradiction, but I'm not sure how to get to it...

Answer 5

the constant sequence is both
but not a very interesting sequence
if the sequence is not constant, then it cannot be both

Answer 6

Well there you go. I was thinking along the more "interesting" lines, I suppose.

Answer 7

There is no contradiction, since there are example of sequences that are both arithmetic and geometric, such as 0,0,... a sequence can be both arithmetic and geometric. However, as satellite said, this isn't a very interesting sequence.

Answer 8

But in general, does my reasoning still work? Given some sequences a_n and A_n.

Answer 9

In general, you would have to show that if \(\large a+sd=ar^s\) and \(a\neq 0\), then \(d=0\) and \(r=1\).

Answer 10

Correction: If \(\large a+sd=ar^s\) for all \(s\in\mathbb{Z}_{\ge0}\), \(a\neq0\), then \(d=0\) and \(r=1\).

Answer 11

Ah, I see now. Thanks!

Answer 12

Also, I only include \(a\neq0\) since if \(a=0\), then all we need is \(d=0\) to guarantee us the sequence \(0,0,...\) (assuming you define \(0^0\)).

Answer 13

If \(a+sd=ar^s\), then for non-zero \(a\), \[\large
\begin{aligned}
\frac{sd}{a}&=r^s-1\\
&=(r-1)(r^{s-1}+r^{s-2}+...+r+1)
\end{aligned}
\]However, both sides must be equal for all values of \(s\ge0\), it must be that both sides are 0. Hence, \(r=1\), and \(sd=0\implies d=0\).