Question

Three positive numbers form an arithmetic progression; their sum is 18. If the first number is increased by 4, then the numbers will form a geometric progression. Find the original three numbers in arithmetic progression.

Answer 1

Answer:

  {4+2√7, 6, 8-2√7}

Step-by-step explanation:

The middle term of three of an arithmetic sequence is the average of the three numbers—their sum divided by 3. So, this arithmetic sequence has a middle term of 6.

If the common difference is x, then the numbers are ...

  {6 -x, 6, 6+x}

When the first is increased by 4, to 10-x, then the ratios of adjacent terms are the same:

  6/(10-x) = (6+x)/6

  36 = 60 +4x -x^2 . . . . . . multiplying by 6(10-x)

  x^2 -4x = 24 . . . . . in a form suitable for completing the square

  (x -2)^2 = 28 . . . . . . after adding 4

  x = 2-2√7 . . . . . the solution that makes the sequence be positive numbers

The arithmetic sequence is then ...

  {4 +2√7, 6, 8 -2√7} ≈ {9.2915, 6, 2.7085}