The sum to infinity of a G.P series is R. The sum to infinity of the squares of the terms is 2R. The sum to infinity of the cubes of the terms is (64/13)R. Find (i) the value of R. (ii) the first term of the first original series.

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choltas5j · Accepted Answer

When none of the terms are changed, let the first term be a and thecommon ratio be r.When the terms are squared, the first term becomes       a    2  and the common ratio becomes       r    2  So,      a          1      −      r        =  RSince            a      2              1      −              r        2              =  2  RThen,            a      2              1      +              r        2              =            2      a              1      −      r                    a      2              (      1      +      r      )      (      1      −      r      )        =            2      a              1      −      r      Removing common terms,      a          1      +      r        =  2a=2r+2If the terms are cubed, the sum is            a      3              1      −              r        3              =      64    13    RSubstituting ,            a      3              1      −              r        3              =  R            a      2              1      −              r        3              =      64    13    (      a          1      −      r        )              a      (              a        2            )              (      1      −      r      )      (      1      +      r      +              r        2            )        =      64    13    (      a          1      −      r        )Removing common terms,            a      2                      r        2            +      r      +      1        =      64    13  Substituting a=2r+2=2(r+1)            8      (              r        2            +      2      r      +      1      )                      r        2            +      r      +      1        =      64    13  Dividing both sides by 8,                    r        2            +      2      r      +      1                      r        2            +      r      +      1        =      8    13  Copyright ©2011-2012 CUI WEI. All Rights Reserved.Cross-multiplying,  13      r    2    +  26  r  +  14  =      8    2    +  8  r  +  8    5      r    2    +  18  r  +  5  =  0After using the quadratic formula, you&#039;ll get r=-3.2967 or-0.3033Since this is a infinite progression values of r with absolutevalues greater than one are not allowed. So, r cannot be-3.2967Given r=-0.3033, a=2(-0.3033)+2&amp;gt;&amp;gt;a=1.393So,R=1.393/(1-(-0.3033))&amp;gt;&amp;gt;R=1.609

the sum to infinity of a G.P series is R. The sum to infinity of the squares of the terms is 2R. The sum to infinity of the cubes of the terms is (64/13)R. Find (i) the value of R. (ii) the first term of the first original series.

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