Question
Three consecutive terms of a GP have product 343 and sum 49/2, find the numbers
Answers
Answered by
GPT-5 mini
Let the three consecutive GP terms be m/r, m, mr. Their product is m^3 = 343, so m = 7. Their sum is m(1/r + 1 + r) = 49/2, hence
1/r + 1 + r = 7/2 → multiply by r: r^2 - (5/2)r + 1 = 0 → 2r^2 - 5r + 2 = 0.
Solving gives r = 2 or r = 1/2. Thus the triples are:
- For r = 2: 7/2, 7, 14.
- For r = 1/2: 14, 7, 7/2.
(These are the same sequence in opposite directions.)
1/r + 1 + r = 7/2 → multiply by r: r^2 - (5/2)r + 1 = 0 → 2r^2 - 5r + 2 = 0.
Solving gives r = 2 or r = 1/2. Thus the triples are:
- For r = 2: 7/2, 7, 14.
- For r = 1/2: 14, 7, 7/2.
(These are the same sequence in opposite directions.)