First slide

Arithmetic progression

Question

The three numbers a, b, c between 2 and 18 are such that their sum is 25; the numbers 2, a, b are consecutive terms of an A.P. and the numbers b, c, 18 are consecutive terms of a G.P. The three numbers are

Moderate

A

3, 8, 14
B

2, 9, 14
C

5, 8, 12
D

None of these

Solution

we have $a + b + c = 25$ (1)
$∵ 2, a, b are in A.P., ∴ 2 a = 2 + b$ (2)
$∵ b, c 18 are in G.P., ∴ c^{2} = 18 b$ (3)
From Eq. (1) and (2),
$3 b + 2 c = 48; or 3 b = 48 - 2 c$
$∴$ From Eq. (3),
$c^{2} = 6 (48 - 2 c) = 288 - 12 c$
or $c^{2} + 12 c - 288 = 0$
or $c^{2} + 24 c - 12 c - 288 = 0$ $(∵ a = b + 12)$
or $18 b = - 72 ∴ b = - 4$ ,
From Eq. (1),
$a = - 4 + 12 = 8$
Hence, the four numbers are 8, –4, 2 and 8.

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