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If the first three terms of an A.P. are ...

If the first three terms of an A.P. are the roots of the equation ` 4x^(3) - 24x^(2) + 23 + 18 = 0` them
compute the sum of the first n terms .

Text Solution

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Let ` alpha - beta , alpha , alpha, beta ` be roots of the given cubic equation ` 4x^(3) - 24 x^(2) + 23 x + 18 - 0 ` them
`alpha - beta + alpha + alpha + beta = 3 alpha = (24)/(4) = 6 rArr alpha = 2 and alpha(alpha -beta)(alpha - beta) = - (18)/(4)`
`rArr alpha^(2) - beta^(2) = - (9)/(4) = 4 - beta^(2)`
` beta^(2) = 4 + (9)/(4) = (25)/(4)`
`rArr beta = pm(5)/(2)`
Thus roots are
` 2- (5)/(2) , 2 + (5)/(2) or 2 + (5)/(2) , 2, 2 - (5)/(2)`
` rArr - (1)/(2) , 2, (9)/(2) or (9)/(2) , 2, - (1)/(2)`
Case I :
When ` - (1)/(2), 2, (9)/(2)` are first three terms of an A.P. them ` a = - (1)/(2) , d = (5)/(2)` and hence
` S_(n)` = Sum to first n terms = ` (n)/(2) ,[ 2a + (n-1)d]`
` = (n)/(2) [ 2xx - (1)/(2) + (n-1) (5)/(2)] = (n (5n-7))/(4)`
Case II : When ` (9)/(2) , 2, - (1)/(2)` are first three terms of an A.P. , them a = ` (9)/(2) , d = - (5)/(2)` and hence
` S_(n)` = Sum to first n terms
` = (n)/(2) [ 2a + (n- 1)d] `
` = (n)/(2) [ 2xx (9)/(2) + (n-1) xx (-5)/(2)]`
` = (n(23 -5n))/(4)`
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