AS Level Maths - Geometric and Arithmetic sequences?

a = a
a + d = ar
a + 2d = ar^2
From the second equation:
ar - a = d
a(r - 1) = d
a = d / (r - 1)
From the third equation:
a - ar^2 = 2d
a(1 - r^2) = 2d
a = (2d) / (1 - r^2)
Combine the above equations:
d / (r - 1) = (2d) / (1 - r^2)
2dr - 2d = d - dr^2
d - dr^2 - 2dr + 2d = 0
3d - 2dr - dr^2 = 0
d(3 - 2r - r^2) = 0
d = 0 or 3 - 2r - r^2 = 0
d = 0 or r^2 + 2r - 3 = 0
In the latter case we have:
r^2 + 3r - r - 3 = 0
r(r + 3) - (r + 3) = 0
(r - 1)(r + 3) = 0
r - 1 = 0 or r + 3 = 0
r = 1 or r = -3

If d = 0 we have:
a = ar
ar - a = 0
a(r - 1) = 0
Since a is not zero, then r - 1 = 0, so r = 1

If r = 1 we have:
a + d = a
a + 2d = a
d = 0

If r = 3 we have:
a + d = 3a, so d = 3a - a = 2a
a + 2d = 9a, so 8a = 2d, so d = 4a
So 2a = 4a, which is impossible because a can't be zero.

(a + 2d) / (a + d) = (a + d) / a
a * (a + 2d) = (a + d) * (a + d)
a^2 + 2ad = a^2 + 2ad + d^2
0 = d^2
0 = d

(a + 2d) / (a + d) =>
(a + 0) / (a + 0) =>
a/a =>
1

So, the common ratio is 1 and the common difference is 0

arithmatic sequence eg a million , 4 , 7,10, 13,... u have a wide-unfold term right here this is a million=a and a distinction btw 2 consecutive words as a effect=3=d sum=n/2(2a+(n-a million)d) n=entire no of words 4-a million=3 = 7-4=.... geometric sequence eg 2,4,8,sixteen,.... first term=2=a(it must be any num) ratio= branch of two consecutive words=2=4/2=8/4=r sum=a( a million-r^n)/(a million-r)