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CBSE Class 11-science Answered

1,5,25 are the pth,qth,and rth terms respectively of G.P .Prove that p,q,r are in A.P .
Asked by Aayushi | 24 Mar, 2017, 10:21: AM
answered-by-expert Expert Answer

begin mathsize 16px style Given space that space 1 comma 5 comma 25 space are space the space pth comma qth comma and space rth space terms space respectively space of space straight G. straight P. Hence comma straight t subscript straight p equals aR to the power of straight n minus 1 end exponent aR to the power of straight p minus 1 end exponent space equals 1............ left parenthesis straight i right parenthesis Also comma straight t subscript straight q equals aR to the power of straight q minus 1 end exponent............ left parenthesis ii right parenthesis straight t subscript straight r equals aR to the power of straight r minus 1 end exponent............. left parenthesis iii right parenthesis After space dividing space left parenthesis straight i right parenthesis space by space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis space by space left parenthesis ii right parenthesis comma straight R to the power of straight q minus straight p end exponent equals 5 space and space straight R to the power of straight r minus straight q end exponent equals 5 space Equating space LHS comma straight q minus straight p equals straight r minus straight q Hence comma space straight p comma straight q comma straight r space are space in space straight A. straight P. end stylebegin mathsize 16px style Given space that space 1 comma 5 comma 25 space are space the space pth comma qth comma and space rth space terms space respectively space of space straight G. straight P. Hence comma straight t subscript straight p equals aR to the power of straight n minus 1 end exponent aR to the power of straight p minus 1 end exponent space equals 1............ left parenthesis straight i right parenthesis Also comma straight t subscript straight q equals aR to the power of straight q minus 1 end exponent............ left parenthesis ii right parenthesis straight t subscript straight r equals aR to the power of straight r minus 1 end exponent............. left parenthesis iii right parenthesis After space dividing space left parenthesis straight i right parenthesis space by space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis space by space left parenthesis ii right parenthesis comma straight R to the power of straight q minus straight p end exponent equals 5 space and space straight R to the power of straight r minus straight q end exponent equals 5 space Equating space LHS comma straight q minus straight p equals straight r minus straight q Hence comma space straight p comma straight q comma straight r space are space in space straight A. straight P. end stylebegin mathsize 16px style Given space that space 1 comma 5 comma 25 space are space the space pth comma qth comma and space rth space terms space respectively space of space straight G. straight P. Hence comma straight t subscript straight p equals aR to the power of straight n minus 1 end exponent aR to the power of straight p minus 1 end exponent space equals 1............ left parenthesis straight i right parenthesis Also comma straight t subscript straight q equals aR to the power of straight q minus 1 end exponent............ left parenthesis ii right parenthesis straight t subscript straight r equals aR to the power of straight r minus 1 end exponent............. left parenthesis iii right parenthesis After space dividing space left parenthesis straight i right parenthesis space by space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis space by space left parenthesis ii right parenthesis comma straight R to the power of straight q minus straight p end exponent equals 5 space and space straight R to the power of straight r minus straight q end exponent equals 5 space Equating space LHS comma straight q minus straight p equals straight r minus straight q Hence comma space straight p comma straight q comma straight r space are space in space straight A. straight P. end stylebegin mathsize 16px style Given space that space 1 comma 5 comma 25 space are space the space pth comma qth comma and space rth space terms space respectively space of space straight G. straight P. Hence comma straight t subscript straight p equals aR to the power of straight n minus 1 end exponent aR to the power of straight p minus 1 end exponent space equals 1............ left parenthesis straight i right parenthesis Also comma straight t subscript straight q equals aR to the power of straight q minus 1 end exponent............ left parenthesis ii right parenthesis straight t subscript straight r equals aR to the power of straight r minus 1 end exponent............. left parenthesis iii right parenthesis After space dividing space left parenthesis straight i right parenthesis space by space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis space by space left parenthesis ii right parenthesis comma straight R to the power of straight q minus straight p end exponent equals 5 space and space straight R to the power of straight r minus straight q end exponent equals 5 space Equating space LHS comma straight q minus straight p equals straight r minus straight q Hence comma space straight p comma straight q comma straight r space are space in space straight A. straight P. end stylebegin mathsize 16px style Given space that space 1 comma 5 comma 25 space are space the space pth comma qth comma and space rth space terms space respectively space of space straight G. straight P. Hence comma straight t subscript straight p equals aR to the power of straight n minus 1 end exponent aR to the power of straight p minus 1 end exponent space equals 1............ left parenthesis straight i right parenthesis Also comma straight t subscript straight q equals aR to the power of straight q minus 1 end exponent............ left parenthesis ii right parenthesis straight t subscript straight r equals aR to the power of straight r minus 1 end exponent............. left parenthesis iii right parenthesis After space dividing space left parenthesis straight i right parenthesis space by space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis space by space left parenthesis ii right parenthesis comma straight R to the power of straight q minus straight p end exponent equals 5 space and space straight R to the power of straight r minus straight q end exponent equals 5 space Equating space LHS comma straight q minus straight p equals straight r minus straight q Hence comma space straight p comma straight q comma straight r space are space in space straight A. straight P. end stylebegin mathsize 16px style Given space that space 1 comma 5 comma 25 space are space the space pth comma qth comma and space rth space terms space respectively space of space straight G. straight P. Hence comma straight t subscript straight p equals aR to the power of straight n minus 1 end exponent aR to the power of straight p minus 1 end exponent space equals 1............ left parenthesis straight i right parenthesis Also comma straight t subscript straight q equals aR to the power of straight q minus 1 end exponent............ left parenthesis ii right parenthesis straight t subscript straight r equals aR to the power of straight r minus 1 end exponent............. left parenthesis iii right parenthesis After space dividing space left parenthesis straight i right parenthesis space by space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis space by space left parenthesis ii right parenthesis comma straight R to the power of straight q minus straight p end exponent equals 5 space and space straight R to the power of straight r minus straight q end exponent equals 5 space Equating space LHS comma straight q minus straight p equals straight r minus straight q Hence comma space straight p comma straight q comma straight r space are space in space straight A. straight P. end stylebegin mathsize 16px style Given space that space 1 comma 5 comma 25 space are space the space pth comma qth comma and space rth space terms space respectively space of space straight G. straight P. Hence comma straight t subscript straight p equals aR to the power of straight n minus 1 end exponent aR to the power of straight p minus 1 end exponent space equals 1............ left parenthesis straight i right parenthesis Also comma straight t subscript straight q equals aR to the power of straight q minus 1 end exponent............ left parenthesis ii right parenthesis straight t subscript straight r equals aR to the power of straight r minus 1 end exponent............. left parenthesis iii right parenthesis After space dividing space left parenthesis straight i right parenthesis space by space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis space by space left parenthesis ii right parenthesis comma straight R to the power of straight q minus straight p end exponent equals 5 space and space straight R to the power of straight r minus straight q end exponent equals 5 space Equating space LHS comma straight q minus straight p equals straight r minus straight q Hence comma space straight p comma straight q comma straight r space are space in space straight A. straight P. end stylebegin mathsize 16px style Given space that space 1 comma 5 comma 25 space are space the space pth comma qth comma and space rth space terms space respectively space of space straight G. straight P. Hence comma straight t subscript straight p equals aR to the power of straight n minus 1 end exponent aR to the power of straight p minus 1 end exponent space equals 1............ left parenthesis straight i right parenthesis Also comma straight t subscript straight q equals aR to the power of straight q minus 1 end exponent............ left parenthesis ii right parenthesis straight t subscript straight r equals aR to the power of straight r minus 1 end exponent............. left parenthesis iii right parenthesis After space dividing space left parenthesis straight i right parenthesis space by space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis space by space left parenthesis ii right parenthesis comma straight R to the power of straight q minus straight p end exponent equals 5 space and space straight R to the power of straight r minus straight q end exponent equals 5 space Equating space LHS comma straight q minus straight p equals straight r minus straight q Hence comma space straight p comma straight q comma straight r space are space in space straight A. straight P. end stylebegin mathsize 16px style Given space that space 1 comma 5 comma 25 space are space the space pth comma qth comma and space rth space terms space respectively space of space straight G. straight P. Hence comma straight t subscript straight p equals aR to the power of straight n minus 1 end exponent aR to the power of straight p minus 1 end exponent space equals 1............ left parenthesis straight i right parenthesis Also comma straight t subscript straight q equals aR to the power of straight q minus 1 end exponent............ left parenthesis ii right parenthesis straight t subscript straight r equals aR to the power of straight r minus 1 end exponent............. left parenthesis iii right parenthesis After space dividing space left parenthesis straight i right parenthesis space by space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis space by space left parenthesis ii right parenthesis comma straight R to the power of straight q minus straight p end exponent equals 5 space and space straight R to the power of straight r minus straight q end exponent equals 5 space Equating space LHS comma straight q minus straight p equals straight r minus straight q Hence comma space straight p comma straight q comma straight r space are space in space straight A. straight P. end stylebegin mathsize 16px style Given space that space 1 comma 5 comma 25 space are space the space pth comma qth comma and space rth space terms space respectively space of space straight G. straight P. Hence comma straight t subscript straight p equals aR to the power of straight n minus 1 end exponent aR to the power of straight p minus 1 end exponent space equals 1............ left parenthesis straight i right parenthesis Also comma straight t subscript straight q equals aR to the power of straight q minus 1 end exponent............ left parenthesis ii right parenthesis straight t subscript straight r equals aR to the power of straight r minus 1 end exponent............. left parenthesis iii right parenthesis After space dividing space left parenthesis straight i right parenthesis space by space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis space by space left parenthesis ii right parenthesis comma straight R to the power of straight q minus straight p end exponent equals 5 space and space straight R to the power of straight r minus straight q end exponent equals 5 space Equating space LHS comma straight q minus straight p equals straight r minus straight q Hence comma space straight p comma straight q comma straight r space are space in space straight A. straight P. end style

Answered by | 24 Mar, 2017, 03:59: PM

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