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# Digits

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A three-digit even number is given such that the hundreds, tens, and unit digits respectively form a decreasing arithmetic sequence. When two is added to the number, the same order of the digits now forms a geometric sequence. What is the original number?

Dec 29, 2021

#1
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A three-digit even number is given such that the hundreds, tens, and unit digits respectively form a decreasing arithmetic sequence. When two is added to the number, the same order of the digits now forms a geometric sequence. What is the original number?

840 will work.  Arithmetic sequence, the difference is –4

842                 Geometric sequence, the multiplier is 1/2

I complain about people who give only the answer, without explaining the steps which led to that answer.

Sorry; I have no steps.  I just contemplated this for a little while and an answer sort of fell out of the sky.

.

Dec 29, 2021
#2
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840       arithmetic series   d = 4

842      geometric series   r = 1/2

840 is the original number

Dec 29, 2021
#3
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Guest Dec 29, 2021
#4
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You DO see they were posted at about the same time. don't you?    The first answer was posted while I was writing my solution and I didn't see it, silly!

BTW   should be    d = - 4

Guest Dec 29, 2021
#5
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Thanks for taking the effort to explain to a silly schlemiel.

Guest Dec 29, 2021
edited by Guest  Dec 29, 2021