Roboguru

Jika , , dan merupakan jumlah suku, suku, dam suku pertama dari deret geometri, buktikan bahwa:

Jika begin mathsize 14px style S subscript 1 end stylebegin mathsize 14px style S subscript 2 end style, dan begin mathsize 14px style S subscript 3 end style merupakan jumlah undefined suku, begin mathsize 14px style 2 n end style suku, dam begin mathsize 14px style 3 n end style suku pertama dari deret geometri, buktikan bahwa:

begin mathsize 14px style S subscript 1 open parentheses S subscript 3 minus S subscript 2 close parentheses equals left parenthesis S subscript 2 minus S subscript 1 right parenthesis squared end style 

  1. undefined 

  2. undefined 

Jawaban:

begin mathsize 14px style S subscript 1 open parentheses S subscript 3 minus S subscript 2 close parentheses equals left parenthesis S subscript 2 minus S subscript 1 right parenthesis squared end style

begin mathsize 14px style S subscript 1 open parentheses S subscript 3 minus S subscript 2 close parentheses equals fraction numerator a open parentheses r to the power of n minus 1 close parentheses over denominator r minus 1 end fraction open square brackets open parentheses fraction numerator a left parenthesis r to the power of 3 n end exponent minus 1 over denominator r minus 1 end fraction close parentheses minus open parentheses fraction numerator a left parenthesis r to the power of 2 n end exponent minus 1 right parenthesis over denominator r minus 1 end fraction close parentheses close square brackets equals fraction numerator a squared open parentheses r to the power of n minus 1 close parentheses open parentheses r to the power of 3 n end exponent minus r to the power of 2 n end exponent close parentheses over denominator left parenthesis r minus 1 right parenthesis squared end fraction equals fraction numerator a squared open parentheses r to the power of 4 n end exponent minus 2 r to the power of 3 n end exponent plus r to the power of 2 n end exponent close parentheses over denominator left parenthesis r minus 1 right parenthesis squared end fraction equals fraction numerator a squared open parentheses r to the power of 2 n end exponent minus r to the power of n close parentheses squared over denominator open parentheses r minus 1 close parentheses squared end fraction equals open parentheses fraction numerator a open parentheses r to the power of 2 n end exponent minus r to the power of n close parentheses over denominator r minus 1 end fraction close parentheses squared equals open parentheses fraction numerator a open parentheses r to the power of 2 n end exponent minus 1 close parentheses over denominator r minus 1 end fraction minus fraction numerator a open parentheses r to the power of n minus 1 close parentheses over denominator r minus 1 end fraction close parentheses squared equals open parentheses S subscript 2 minus S subscript 1 close parentheses squared end style 

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