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Hi guys this is my first time using the forums and I need help on 1 question:

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 + ....]

I know to start the problem you somehow rewrite these infinite series but idk how. Please help I love you guys!

 Feb 22, 2020
 #1
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Using the quadratic equation to find the solutions to  7x2 -x - 3  =  0, we can get these answers:

a  =  [ 1 + sqrt(85) ] / 14       and     b  =  [ 1 - sqrt(85) ] / 14 

 

Both  1 + a + a2 + a3 + ...     and    1 + b + b2 + b3​ + ...     

are  infinite geometric series with common ratios in the range  -1 < r < 1,

so we can use the formula:  Sum  =  a1 / ( 1 - r)

 

In both cases,  a1 = 1, so:

the sum of  1 + a + a2 + a3 + ...  is  1 / [ 1 - (1 + sqrt(85))/14 ]  

     =  14 / [ 14 - (1 + sqrt(85) ]     =     14 / [ 13 - sqrt(85) ]

 

and the sum of 1 + b + b2 + b3​ + ...   is  14 / [ 13 + sqrt(85) ]

 

Multiplying these two values together, we get  196 / (169 - 85)   196 / 84  =  7/3

 Feb 22, 2020
 #2
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Thank you so much!!!

Guest Feb 23, 2020

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