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CBSE Class 10 Answered

Given, In an Arithmetic Progression, the p to the power of t h end exponentterm is equal to 1 over qand the q to the power of t h end exponentterm is equal to 1 over p. Prove that a subscript p plus q end subscript equals 1 over q plus 1 over p
Asked by Ra. | 17 Dec, 2015, 10:23: PM
answered-by-expert Expert Answer
L e t space space a subscript 1 space b e space t h e space i n i t i a l space t e r m space a n d space d space b e space t h e space c o m m o n space d i f f e r e n c e space o f space t h e space g i v e n space A P G i v e n space t h a t space p t h space t e r m equals space 1 over q rightwards double arrow a subscript p equals a subscript 1 plus open parentheses p minus 1 close parentheses d equals 1 over q... open parentheses 1 close parentheses A n d space q t h space t e r m equals 1 over p rightwards double arrow a subscript q equals a subscript 1 plus open parentheses q minus 1 close parentheses d equals 1 over p... open parentheses 2 close parentheses S u b t r a c t i n g space e q u a t i o n s space open parentheses 1 close parentheses space f r o m space open parentheses 2 close parentheses comma space w e space h a v e comma 1 over q minus 1 over p equals p d minus q d rightwards double arrow fraction numerator p minus q over denominator p q end fraction equals open parentheses p minus q close parentheses d rightwards double arrow fraction numerator 1 over denominator p q end fraction equals d... left parenthesis 3 right parenthesis S u b s t i t u t i n g space t h e space v a l u e space o f space d space f r o m space e q u a t i o n space left parenthesis 3 right parenthesis space i n space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e comma a subscript 1 plus open parentheses p minus 1 close parentheses fraction numerator 1 over denominator p q end fraction equals 1 over q rightwards double arrow a subscript 1 plus fraction numerator p over denominator p q end fraction minus fraction numerator 1 over denominator p q end fraction equals 1 over q rightwards double arrow a subscript 1 plus 1 over q minus fraction numerator 1 over denominator p q end fraction equals 1 over q rightwards double arrow a subscript 1 minus fraction numerator 1 over denominator p q end fraction equals 0 rightwards double arrow a subscript 1 equals fraction numerator 1 over denominator p q end fraction... left parenthesis 4 right parenthesis L e t space u s space c o n s i d e r space t h e space open parentheses p plus q minus 1 close parentheses t h space t e r m space o f space t h e space g i v e n space A P a subscript p plus q end subscript equals a subscript 1 plus open parentheses q minus 1 close parentheses d S u b s t i t u t i n g space t h e space v a l u e space o f space a space a n d space d space f r o m space e q u a t i o n s space left parenthesis 4 right parenthesis space a n d space left parenthesis 3 right parenthesis space i n space t h e space a b o v e space e q u a t i o n comma space w e space h a v e comma a subscript p plus q end subscript equals fraction numerator 1 over denominator p q end fraction plus open parentheses p plus q minus 1 close parentheses fraction numerator 1 over denominator p q end fraction rightwards double arrow a subscript p plus q end subscript equals fraction numerator 1 over denominator p q end fraction plus fraction numerator p over denominator p q end fraction plus fraction numerator q over denominator p q end fraction minus fraction numerator 1 over denominator p q end fraction rightwards double arrow a subscript p plus q end subscript equals 1 over p plus 1 over q H e n c e space p r o v e d.
Answered by Vimala Ramamurthy | 18 Dec, 2015, 10:15: AM

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