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Let a and b be the roots of 7x^2 - x - 3 = 0. Find (1 + a + a^2 + a^3 +.....) (1 + b + b^2 + b^3 +.....).

 

Thx :)

 Feb 26, 2022
 #1
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Hint:

1)  Use the quadratic formula to find the two solutions to the equation:  7x2 - x - 3  =  0.

      Leave them in reduced radical form.

      Label one solution as 'a' and the other as 'b'.

      Note that each solution is in the range:  -1 < x < 1,

 

2)  1 + a + a2 + a3 + ...  is an infinite geometric series, whose sum can be found

     using the formula:  Sum  =  (first term) / [ 1 - (common ratio) ]

     the first term is 1, and the common ratio is the solution that you found in step 1.

     This formula applies because the common ration is in the range -1 < r < 1.

 

3)  You can find the sum of 1 +  + b2 + b3 + ...   in a similar way and multiply these

     two answers.

 Feb 26, 2022

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