Three integers are in the ratio 2:3:8. If 4 is added to the middle number, the resulting number is the second term of a geometric progression of which the other two integers are the first and third terms. How do you find the three integers?

Mariyah Bell

Mariyah Bell

Answered question

2022-10-26

Three integers are in the ratio 2:3:8. If 4 is added to the middle number, the resulting number is the second term of a geometric progression of which the other two integers are the first and third terms. How do you find the three integers?

Answer & Explanation

Layne Murillo

Layne Murillo

Beginner2022-10-27Added 14 answers

If the first term is 2x then the sequence formed by adding 4 to the middle number is:
2x,3x+4,8x
The middle of three terms of a geometric sequence is equal to the geometric mean of the preceding and following terms, so:
3 x + 4 = ± 2 x 8 x = ± 16 x 2 = ± 4 x
If 3x+4=4x then x=4 and our original integers are 8, 12, and 32.
The geometric sequence is 8, 16, 32 with common ratio 2.
If 3x+4=−4x then x = - 4 7 , but this is not an integer and does not give rise to integers when multiplied by 2, 3 and 8.
Out of curiosity let's look at this alternative non-integer solution:
2 x = - 8 7
3 x + 4 = - 12 7 + 4 = 16 7
8 x = - 32 7
So the common ratio of this geometric sequence is −2 as expected.

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