If 2, 7, 9, 5 are subtracted respectively from four
numbers forming a G.P., the resulting numbers are
in A.P., then the smallest of the four numbers is
Answers
numbers are ; -3, -6, -12, -24
let a , ar, ar² , ar³ are four number in GP.
a/c to question,
(a - 2), (ar - 7), (ar² - 9), (ar³ - 5) are in AP.
so, common difference remains constant.
i.e., (ar - 7) - (a - 2) = (ar² - 9) - (ar - 7)
⇒ar - a - 5 = ar² - ar - 2
⇒ar² - 2ar + a = -3 ......(1)
again, (ar² - 9) - (ar - 7) = (ar³ - 5) - (ar² - 9)
⇒ar² - ar - 2 = ar³ - ar² + 4
⇒ar³ - 2ar² + ar + 6 = 0
⇒r(ar² - 2ar + a) + 6 = 0
from equation (1),
⇒r(-3) + 6 = 0.
r = 2
so, a(2)² - 2a(2) + a = -3
⇒4a - 4a + a = -3
⇒a = -3
now, four numbers are ;
a = -3, ar = -3(2) = -6,
ar² = -3(2)² = -12
and ar³ = -3(2)³ = -24
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