Permutations and Combinations

Permutations and Combinations

Question

Find the sum to indicated number of terms in each of the geometric progressions:

1, - a, straight a squared comma spacenegative straight a cubed comma space space space..... semicolon space n terms left parenthesis straight a not equal to space minus 1 right parenthesis

Answer

The given G.P. is 1, -a, a2, -a3, ...........
Here, first term A = 1, straight R space equals space fraction numerator negative straight a over denominator 1 end fraction space equals space minus straight a
                        space space straight S subscript straight n space equals space straight A space fraction numerator left parenthesis straight R to the power of straight n minus 1 right parenthesis over denominator straight R minus 1 end fraction space equals space fraction numerator 1 left square bracket left parenthesis negative straight a right parenthesis to the power of straight n minus 1 right square bracket over denominator negative straight a minus 1 end fraction space equals space fraction numerator 1 minus left parenthesis negative straight a right parenthesis to the power of straight n over denominator 1 plus straight a end fraction

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