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Find three numbers a,b,c between 2 & 18 ...

Find three numbers a,b,c between 2 & 18 such that; (G) their sum is 25 (ii) the numbers 2,a,b are consecutive terms of an AP & (ii) the numbers b,c, 18 are consecutive terms of a G.P.

Text Solution

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The correct Answer is:
`(a = 5) (b = 8) (c = 12)`
If a, b, c `in (2, 18)`, then
`a + b + c = 25`...(i)
Since, 2, a, b are in AP
`rArr 2a = b +2`...(ii)
and b, c 18 are in GP
`rArr c^(2) = 18b`..(iii) ltbr. From Eqs. (i), (ii) and (iii)
`(b+2)/(2) + b + sqrt(18b) = 25`
`rArr 3b + 2 + 6 sqrt2 sqrtb = 50`
`rArr 3b + 6 sqrt2 sqrtb - 48 = 0`
`rArr b + 2 sqrt2 sqrtb - 16 = 0`
`rArr b + 4 sqrt2 sqrtb - 2 sqrt2 sqrtb - 16 = 0`
`rArr sqrtb (sqrtb + 4 sqrt2) - 2 sqrt2 (sqrtb + 4 sqrt2) = 0`
`rArr (sqrtb - 2sqrt2) (sqrtb + 4 sqrt2) = 0`
`rArr b = 8, a = 5`
and `c = 12`
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