SOLUTION: The second, fourth and eight terms of an A.P are in geometrical progression and the sum of the third and fifth term is 20. Find the first four terms of the progression.

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Question 1153525: The second, fourth and eight terms of an A.P are in geometrical progression and the sum of the third and fifth term is 20. Find the first four terms of the progression.
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

AP:
2nd term: a+%2B+d,
4th term: a+%2B+3d,
8th term: a+%2B+7d
These 3 terms are in GP:
+++%28a+%2B+d%29%28a+%2B+7d%29++-%28a+%2B+3d%29%5E2+=0
a%5E2+%2B+8+a+d+%2B+7+d%5E2-%28a%5E2+%2B+6+a+d+%2B+9+d%5E2%29=0
a%5E2+%2B+8+a+d+%2B+7+d%5E2-a%5E2+-+6+a+d+-+9+d%5E2=0
+2a+d++-+2+d%5E2=0
2d%28a+-+d%29+=+0
2ad+-+2d%5E2=0+
2ad+=2d%5E2+
a+=+d



Sum of 3rd and 5th terms is:

%28a+%2B+2d%29+%2B+%28a+%2B+4d%29+=20
2a+%2B+6d+=+20
++8d+=+20
d+=+2.5 or d=%285%2F2%29
=>+a+=+2.5or %285%2F2%29
AP:
1st term: 2.5
2nd term: 5
3rdterm:7.5
4th term: 10
5thterm:12.5
8th term: 20

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


If the sum of the third and fifth terms of the AP is 20, then the 4th term is 10. (Any term in an AP is the average of the two terms before and after it.) So

2nd term: 10-2d
4th term: 10
8th term: 10+4d

The 2nd, 4th, and 8th terms form a GP:

%2810%2B4d%29%2F10+=+10%2F%2810-2d%29
100+=+100%2B20d-8d%5E2
8d%5E2-20d+=+0
8d%28d-2.5%29+=+0
d+=+0 or d+=+2.5

The common difference in the AP is either 0 or 2.5. An AP with common difference 0 is not very interesting, so we can assume the common difference is 2.5.

Then, with common difference 2.5 and 4th term 10, the first four terms are

2.5, 5, 7.5, 10