The sum of the sequence is 129.
The increasing sequence has four positive integers.
The first term of the sequence form an arithmetic sequence.
The last three terms of the sequence form a geometric progression.
The difference between the first and fourth term of the sequence is 30.
So, the sequence will be:
a − d, a, a + d, a − d + 30
As the last three terms are in GP,
( a + d )² = a( a - d + 30 )
a² + 2ad + d² = a² - ad + 30a
3ad = 30a - d
3a = d² / ( 10 - d)
Hence, the right-hand side must be a multiple of 3,
When d = 3, RHS = 9/7 not a multiple of 3.
When, d = 6, RHS = 9.
So, a = 3, which makes a - d negative. So not possible.
Now when,
d = 9, RHS is 81.
a = 27
Therefore, the sequence is:
18, 27, 36, 48.
The sum of the sequence is 129.
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