Let the arithmetic sequence be defined by a_n = a + knan=a+kn for n = 0, 1, 2,...
Let the geometric sequence be defined by b_n = b*r^n for n = 0, 1, 2,...
Then the result of adding these sequences has general term:
c_n = a + kn + b*r^n
Arithmetic sequence ?
c_2 - c_1 = (a+2k+br^2)-(a+k+br) = k + b(r^2-r)
= k + br(r-1)
c_1 - c_0 = (a+k+br)-(a+b) = k + b(r - 1)
If this is an arithmetic sequence, then these two expressions must be equal, so:
color(red)(cancel(color(black)(k))) + br(r-1) = color(red)(cancel(color(black)(k))) + b(r-1)
Hence b = 0 or r = 1 (or both).
That means that the geometric sequence b_0, b_1, b_2,... must be constant.
Geometric sequence ?
If b != 0 and abs(r) > 1 then c_(n+1)/c_n -> r as n -> oo.
If c_0, c_1, c_2,... is a geometric sequence then the ratio between successive terms must be constant, so we require c_1 / c_0 = r
r = c_1 / c_0 = (a+k+br)/(a+b)
Multiply both ends by (a+b) to find:
a+k+color(red)(cancel(color(black)(br))) = ar+color(red)(cancel(color(black)(br)))
Also
r = c_2 / c_1 = (a+2k+br^2)/(a+k+br)
Multiply both ends by (a+k+br) to get:
ar+kr+color(red)(cancel(color(black)(br^2))) = a+2k+color(red)(cancel(color(black)(br^2)))
Hence we have a+k = ar and a+2k = ar^2
So:
a^2 + 2ak = a(a+2k) = a^2r^2 = (ar)^2 = (a+k)^2
=a^2+2ak+k^2
So k = 0
Then ar = a + k = a, so a = 0 or r = 1
Well we assumed abs(r) > 1 so r != 1, so a = 0
If a=0 and k=0 then the arithmetic sequence is constant 0.
Now consider the case abs(r) <= 1.
If the arithmetic sequence is non-constant, then c_(n+1)/c_n -> 1 as n -> oo. Hence we require the common ratio of the geometric sequence to be 1. Then it's a constant sequence, therefore arithmetic and we require b=0 or r = 1 (or both). As a result we also require the arithmetic sequence to be constant, contradicting our supposition.
