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Jika , , ....,  merupakan deret geometri dengan , ...

Jika begin mathsize 14px style U subscript 1 end stylebegin mathsize 14px style U subscript 2 end style, ...., undefined merupakan deret geometri dengan begin mathsize 14px style r not equal to 1 end style

begin mathsize 14px style S equals U subscript 1 plus U subscript 2 plus U subscript 3 plus. space. space. space plus U subscript n end style dan

begin mathsize 14px style T equals 1 over U subscript 1 plus 1 over U subscript 2 plus. space. space. space plus 1 over U subscript n end style 

Buktikan bahwa: begin mathsize 14px style S over T equals U subscript 1 space. space U subscript n end style 

  1. undefined 

  2. undefined 

Jawaban:

Diketahui: 

Deret Geometri

begin mathsize 14px style S equals U subscript 1 plus U subscript 2 plus U subscript 3 plus. space. space. space plus U subscript n end style

begin mathsize 14px style T equals 1 over U subscript 1 plus 1 over U subscript 2 plus. space. space. space plus 1 over U subscript n end style

Buktikan bahwa: begin mathsize 14px style S over T equals U subscript 1 space. space U subscript n end style  

Jawab: 

begin mathsize 14px style S over T equals fraction numerator a plus a r plus a r squared plus... plus a r to the power of n minus 1 end exponent over denominator begin display style 1 over a end style plus begin display style fraction numerator 1 over denominator a r end fraction end style plus begin display style fraction numerator 1 over denominator a r squared end fraction end style plus... plus begin display style fraction numerator 1 over denominator a r to the power of n minus 1 end exponent end fraction end style end fraction S over T equals fraction numerator a open parentheses 1 plus r plus r squared plus... plus r to the power of n minus 1 end exponent close parentheses over denominator begin display style 1 over a end style open parentheses begin display style 1 over 1 end style plus begin display style 1 over r end style plus begin display style 1 over r squared end style plus... plus begin display style 1 over r to the power of n minus 1 end exponent end style close parentheses end fraction S over T equals a squared. fraction numerator open parentheses 1 plus r plus r plus... plus r to the power of n minus 1 end exponent close parentheses over denominator open parentheses begin display style fraction numerator r to the power of n minus 1 end exponent plus... plus r squared plus r plus 1 over denominator r to the power of n minus 1 end exponent end fraction end style close parentheses end fraction equals a squared times r to the power of n minus 1 end exponent S over T equals a times a r to the power of n minus 1 end exponent S over T equals U subscript 1 times U subscript n end style 

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