Archive
24th of March 2017
Mathematics
Q:

1,5,25 are the pth,qth,and rth terms respectively of G.P .Prove that p,q,r are in A.P .

Aayushi Mahida - CBSE - Class XI Science

Friday, March 24, 2017 at 10:21:AM

A:

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

Friday, March 24, 2017 at 15:59:PM