calculate the value of r and a from systems-of-equations given below? a ( 1 − r 7 ) 1 − r = 86 a ( 1 − r 10 ) 1 − r = − 682

Question

calculate the value of   r and a from systems-of-equations given below?            a      (      1      −              r        7            )              1      −      r        =  86            a      (      1      −              r              10            )              1      −      r        =  −  682

Meadow Knox · Accepted Answer

Hint: We have  a  (  1  −      r    7    )  =  86  (  1  −  r  )and  a  (  1  −      r      10    )  =  −  682  (  1  −  r  )Dividing both equations, simplifying and factorizing we get  −  2  (  r  −  1  )  (  r  +  2  )      (    43          r      8        −    43          r      7        +    129          r      6        +    126          r      5        +    132          r      4        +    120          r      3        +    144          r      2        +    96    r    +    192    )    =  0

Xiomara Poole · Accepted Answer

Make the ratio of the two expressions to get            1      −              r              10                    1      −              r        7              =  −      341    43  In absolute value, the rhs is large and then r can be &quot;large&quot;; so may be, for the time being we could write            1      −              r              10                    1      −              r        7              =  −      341    43    ≃            r          10              r      7        =      r    3  Use you calculator to get, as an approximation      r    0    =  −            341      43        3    ≈  −  1.99417If you dont want to jumpt to the obvious conclusion, now consider that you look for the zero of function  f  (  r  )  =  43  (  1  −      r      10    )  +  341  (  1  −      r    7    )and build the Taylor series around   r  =      r    0  ; this would give  f  (  r  )  =      (    −    43          r      0          10        −    341          r      0      7        +    384    )    −      (    430          r      0      9        +    2387          r      0      6        )    (  r  −      r    0    )  +  O      (    (    r    −          r      0              )      2        )  Ignoring the higher order tems, solve for   r to get  r  =      r    0    −            43              r        0              10            +      341              r        0        7            −      384                      r        0        6                    (        430                  r          0          3                +        2387        )            You know the exact value of       r    0    3  , then the exact value of       r    0    6  ; now approximate       r    0    7    ≃  −  2      r    0    6   and       r    0      10    ≃  −  2      r    0    3          r    0    6  . This then gives      r    1    =      r    0    −      236672    39651821    =  −  2.00014You do not need to solve a polynomial of degree   10.

calculate the value of r and a from systems-of-equations given below? a (

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