Does there exist an arithmetic progression, with a positive first term a and a positive common difference d such that the first three terms of the arithmetic progression are also the first three terms of a geometric progression?
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Simple and nice! Well done!
Now, let us assume that the first 3 terms of an AP are also the first 3 terms of a GP.
The AP: a , a + d , a + 2 d , …
The GP: a , a r , a r 2 , …
Since the first 3 terms are the same,
a + d = a r ⟹ Eq.(1)
a + 2 d = a r 2 ⟹ Eq.(2)
Eq.(1) × 2 − Eq.(2):
2 ( a + d ) − ( a + 2 d ) = 2 a r − a r 2 a = 2 a r − a r 2
Since a > 0 , we can divide the equation by a :
1 = 2 r − r 2 r 2 − 2 r + 1 = 0 ( r − 1 ) 2 = 0 r − 1 = 0 r = 1
For the first 3 terms of an AP and GP to be the same, r must be 1 . However,
a + d = a ( 1 ) d = a − a = 0
If r = 1 , d = 0
This means that for an AP and GP to share the same first 3 terms, the common difference must be 0 .
This in turn implies that for an AP with a positive common difference, the first 3 terms of the sequence cannot be the first 3 terms of a GP.
The answer is No
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I think an easier solution is to treat the first three terms of an AP as if they are a GP:
a a + d ∴ ( a + d ) 2 = a + d a + 2 d = a ( a + 2 d )
Simplifying this gives d 2 = 0 ⇒ d = 0 , which is not positive.
Therefore there is no solution.