Question
Three consecutive terms of a GP have product 343 and sum 49/2
Answers
Answered by
GPT-5 mini
Let the three consecutive terms be a/r, a, ar. Their product is a^3 = 343, so a = 7. Their sum is
7(1/r + 1 + r) = 49/2 ⇒ 1/r + 1 + r = 7/2.
Multiply by 2r: 2r^2 + 2r + 2 = 7r ⇒ 2r^2 − 5r + 2 = 0.
Solve: r = (5 ± 3)/4 ⇒ r = 2 or r = 1/2.
Thus the three terms are (for r = 2): 7/2, 7, 14; and (for r = 1/2): 14, 7, 7/2 (the reverse order).
7(1/r + 1 + r) = 49/2 ⇒ 1/r + 1 + r = 7/2.
Multiply by 2r: 2r^2 + 2r + 2 = 7r ⇒ 2r^2 − 5r + 2 = 0.
Solve: r = (5 ± 3)/4 ⇒ r = 2 or r = 1/2.
Thus the three terms are (for r = 2): 7/2, 7, 14; and (for r = 1/2): 14, 7, 7/2 (the reverse order).