Q

Asked by araima2001 | 3rd Sep, 2016, 08:07: AM

Expert Answer:

begin mathsize 12px style Let space the space first space term space of space AP space be space straight A space and space common space difference space be space straight d.
Sum space of space the space first space straight p space terms space is space straight S subscript straight p equals straight p over 2 left square bracket 2 straight A plus left parenthesis straight p minus 1 right parenthesis straight d right square bracket
that space is comma space straight a equals straight p over 2 left square bracket 2 straight A plus left parenthesis straight p minus 1 right parenthesis straight d right square bracket.... left parenthesis straight i right parenthesis
Sum space of space the space first space straight q space terms space is space straight S subscript straight q equals straight q over 2 left square bracket 2 straight A plus left parenthesis straight q minus 1 right parenthesis straight d right square bracket
that space is comma space straight b equals straight q over 2 left square bracket 2 straight A plus left parenthesis straight q minus 1 right parenthesis straight d right square bracket.... left parenthesis ii right parenthesis
Sum space of space the space first space straight r space terms space is space straight S subscript straight r equals straight r over 2 left square bracket straight A plus left parenthesis straight r minus 1 right parenthesis straight d right square bracket
that space is comma space straight c equals straight r over 2 left square bracket 2 straight A plus left parenthesis straight r minus 1 right parenthesis straight d right square bracket.... left parenthesis iii right parenthesis
Multiplying space left parenthesis straight i right parenthesis space by space fraction numerator straight q minus straight r over denominator straight p end fraction comma space left parenthesis ii right parenthesis space by space fraction numerator straight r minus straight p over denominator straight q end fraction comma space left parenthesis iii right parenthesis space by space fraction numerator straight p minus straight q over denominator straight r end fraction space and space adding space them comma space we space get
straight a over straight p left parenthesis straight q minus straight r right parenthesis plus straight b over straight q left parenthesis straight r minus straight p right parenthesis plus straight c over straight r left parenthesis straight p minus straight q right parenthesis
equals left parenthesis straight q minus straight r right parenthesis 1 half left square bracket 2 straight A plus left parenthesis straight p minus 1 right parenthesis straight d right square bracket plus left parenthesis straight r minus straight p right parenthesis 1 half left square bracket 2 straight A plus left parenthesis straight q minus 1 right parenthesis straight d right square bracket plus left parenthesis straight p minus straight q right parenthesis 1 half left square bracket 2 straight A plus left parenthesis straight r minus 1 right parenthesis straight d right square bracket
equals left parenthesis straight q minus straight r right parenthesis 1 half 2 straight A plus left parenthesis straight q minus straight r right parenthesis 1 half left parenthesis straight p minus 1 right parenthesis straight d plus left parenthesis straight r minus straight p right parenthesis 1 half 2 straight A plus left parenthesis straight r minus straight p right parenthesis 1 half left parenthesis straight q minus 1 right parenthesis straight d plus left parenthesis straight p minus straight q right parenthesis 1 half 2 straight A plus left parenthesis straight p minus straight q right parenthesis 1 half left parenthesis straight r minus 1 right parenthesis straight d right square bracket
equals fraction numerator 2 straight A over denominator 2 end fraction left square bracket straight q minus straight r plus straight r minus straight p plus straight p minus straight q right square bracket plus straight d over 2 left square bracket left parenthesis straight q minus straight r right parenthesis left parenthesis straight p minus 1 right parenthesis plus left parenthesis straight r minus straight p right parenthesis left parenthesis straight q minus 1 right parenthesis plus left parenthesis straight p minus straight q right parenthesis left parenthesis straight r minus 1 right parenthesis right square bracket
equals fraction numerator 2 straight A over denominator 2 end fraction left square bracket 0 right square bracket plus straight d over 2 left square bracket pq minus straight q minus rp plus straight r plus rq minus straight r minus pq plus straight p plus pr minus straight p minus rq plus straight q right square bracket
equals fraction numerator 2 straight A over denominator 2 end fraction left square bracket 0 right square bracket plus straight d over 2 left square bracket 0 right square bracket
equals 0

end style

Answered by Rebecca Fernandes | 3rd Sep, 2016, 06:35: PM