Permutations and Combinations

Permutations and Combinations

Question

In a finite GP. the product of the terms equidistant from the beginning and the end is always same and equal to the product of first and last term.

Answer

Let straight t subscript 1 comma space space straight t subscript 2 comma space straight t subscript 3 comma......... space space straight t subscript straight n minus 1 end subscript comma space straight t subscript straight n be a finite G.P. with common ratio r. Then,
                   straight t subscript straight m = mth term from the beginning = straight t subscript 1 space straight r to the power of straight m minus 1 end exponent
                   space space straight t subscript straight n space equals space straight m space plus space 1 space equals space mth space term space from space the space end space equals space straight t subscript straight n open parentheses 1 over straight r close parentheses to the power of straight m minus 1 end exponent comma space space space where space 1 space less than space straight m space less than space straight n
Now,       straight t subscript straight m. space straight t subscript straight n minus straight m plus 1 end subscript space equals space straight t subscript 1. space straight r to the power of straight m minus 1 end exponent. space straight t subscript straight n open parentheses 1 over straight r close parentheses to the power of straight m minus 1 end exponent space equals space straight t subscript 1 space straight t subscript straight n space horizontal strike logical or 1 space less than space straight m space less than straight n

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