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Binomial Theorem

Question
CBSEENMA11015620
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If a1, a2, … , an are in H.P., then the expression a1a2 + a2a3 + … + an−1an is equal to

  • n(a1 − an)

  • (n − 1) (a1 − an)

  • na1an

  • (n − 1)a1an

Solution

D.

(n − 1)a1an

1 over straight a subscript 2 space minus 1 over straight a subscript 1 space equals 1 over straight a subscript 3 minus 1 over straight a subscript 2 space equals space........ space equals 1 over straight a subscript straight n minus 1 over straight a subscript straight n minus 1 end subscript space equals straight d Then space straight a subscript 1 straight a subscript 2 space equals space fraction numerator straight a subscript 1 minus straight a subscript 2 over denominator straight d end fraction comma space straight a subscript 2 straight a subscript 3 space equals space fraction numerator straight a subscript 2 minus straight a subscript 3 over denominator straight d end fraction comma........ comma straight a subscript straight n minus 1 end subscript straight a subscript straight n space equals space fraction numerator straight a subscript straight n minus 1 end subscript minus straight a subscript straight n over denominator straight d end fraction therefore space straight a subscript 1 straight a subscript 2 space plus space straight a subscript 2 straight a subscript 3 space plus......... space plus straight a subscript straight n minus 1 end subscript straight a subscript straight n space equals space fraction numerator straight a subscript 1 minus straight a subscript straight n over denominator straight d end fraction space Also comma space 1 over straight a subscript straight n space equals 1 over straight a subscript 1 space plus space left parenthesis straight n minus 1 right parenthesis straight d rightwards double arrow space fraction numerator straight a subscript 1 minus straight a subscript straight n over denominator straight d end fraction space equals space left parenthesis straight n minus 1 right parenthesis straight a subscript 1 straight a subscript straight n

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