標題:
geometric Qeustion
發問:
Three numbers x, y and z are in the ratio 2 : 5 : 8. If 6, 15 and 49 are added to x, y and z respectively, they form a geometric sequence. Find the values of z, y and z. 更新: why x = -6, y = -15 and z = -24 ( rejected )
Suppose x = 2k, y = 5k and z = 8k where k is a constant. Since after addition, the numbers form a G.P., so ( y + 15 ) / ( x + 6 ) = ( z + 49 ) / ( y + 15 ) ( y + 15 )^2 = ( x + 6 )( z + 49 ) ( 5k + 15 )^2 = ( 2k + 6 )( 8k + 49 ) 25k^2 + 150k + 225 = 16k^2 + 146k + 294 9k^2 + 4k - 69 = 0 ( 9k - 23 )( k + 3 ) = 0 k = 23 / 9 or -3 When k = 23 / 9, x = 46 / 9, y = 115 / 9 and z = 184 / 9 When k = -3, x = -6, y = -15 and z = -24 ( rejected ) Since the no. will become 0,0 and 25 after addition 2007-11-14 21:25:49 補充： Because the no. will become 0,0 and 25 after addition and this is not a G.P.
其他解答:
geometric Qeustion
發問:
Three numbers x, y and z are in the ratio 2 : 5 : 8. If 6, 15 and 49 are added to x, y and z respectively, they form a geometric sequence. Find the values of z, y and z. 更新: why x = -6, y = -15 and z = -24 ( rejected )
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最佳解答:Suppose x = 2k, y = 5k and z = 8k where k is a constant. Since after addition, the numbers form a G.P., so ( y + 15 ) / ( x + 6 ) = ( z + 49 ) / ( y + 15 ) ( y + 15 )^2 = ( x + 6 )( z + 49 ) ( 5k + 15 )^2 = ( 2k + 6 )( 8k + 49 ) 25k^2 + 150k + 225 = 16k^2 + 146k + 294 9k^2 + 4k - 69 = 0 ( 9k - 23 )( k + 3 ) = 0 k = 23 / 9 or -3 When k = 23 / 9, x = 46 / 9, y = 115 / 9 and z = 184 / 9 When k = -3, x = -6, y = -15 and z = -24 ( rejected ) Since the no. will become 0,0 and 25 after addition 2007-11-14 21:25:49 補充： Because the no. will become 0,0 and 25 after addition and this is not a G.P.
其他解答:
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