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Nth Term of a GP

Which term of...

Question

Which term of the G.P. :

$- 10, \frac{5}{\sqrt{3}}, - \frac{5}{6}, . . . . . . . . . . i s - \frac{5}{72} ?$

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Solution

First term, a = -10

Common ratio ( r ) =

$r equals open parentheses bevelled fraction numerator fraction numerator 5 over denominator square root of 3 end fraction over denominator negative 10 space end fraction close parentheses space space equals negative fraction numerator 1 over denominator 2 square root of 3 end fraction$

$negative 10 cross times open parentheses negative fraction numerator 1 over denominator 2 square root of 3 end fraction close parentheses to the power of open parentheses n minus 1 close parentheses end exponent equals negative 5 over 72$

$open parentheses negative fraction numerator 1 over denominator 2 square root of 3 end fraction close parentheses to the power of open parentheses n minus 1 close parentheses end exponent equals fraction numerator 1 over denominator 2 cross times 72 end fraction open parentheses negative fraction numerator 1 over denominator 2 square root of 3 end fraction close parentheses to the power of open parentheses n close parentheses end exponent equals open parentheses fraction numerator 1 over denominator 2 cross times 72 end fraction close parentheses cross times open parentheses negative fraction numerator 1 over denominator 2 square root of 3 end fraction close parentheses open parentheses negative fraction numerator 1 over denominator 2 square root of 3 end fraction close parentheses to the power of open parentheses n close parentheses end exponent equals open parentheses negative fraction numerator 1 over denominator 4 cross times 72 cross times square root of 3 end fraction close parentheses open parentheses negative fraction numerator 1 over denominator 2 square root of 3 end fraction close parentheses to the power of open parentheses n close parentheses end exponent equals open parentheses negative fraction numerator 1 over denominator 2 to the power of 5 cross times open parentheses square root of 3 close parentheses to the power of 5 end fraction close parentheses open parentheses negative fraction numerator 1 over denominator 2 square root of 3 end fraction close parentheses to the power of open parentheses n close parentheses end exponent equals open parentheses negative fraction numerator 1 over denominator 2 to the power of blank square root of 3 end fraction close parentheses to the power of 5 s o comma space n equals 5$

Therefore ,

5th term in given G. P =

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