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Ques) Let a 1 ,a 2 ,a 3 ,..... be in A.P and a p , a q , a r be in G.P. Then show that ap / aq = (q-p) / (r - q).
since, ap,aq & ar are d terms taken from the given A.P. so, ap = aq - (q-p)*d ..............................(1) aq = ar - (r-q)*d..........................(2) now since ap,aq & ar are in G.P. so aq = R*ap and, ar = R*aq, where R is d common ratio of tge G.P. we put aq = R*ap in (1) we get ap( R-1) = (q-p)d ............................................(3) we put ar = R* aq in (2) we get aq (R-1) = (r-p) d .............................................(4) dividing (3) by (4) we get ap/aq = (q-p)/ (r-p)
since, ap,aq & ar are d terms taken from the given A.P.
so, ap = aq - (q-p)*d ..............................(1)
aq = ar - (r-q)*d..........................(2)
now since ap,aq & ar are in G.P. so aq = R*ap
and, ar = R*aq, where R is d common ratio of tge G.P.
we put aq = R*ap in (1) we get
ap( R-1) = (q-p)d ............................................(3)
we put ar = R* aq in (2) we get
aq (R-1) = (r-p) d .............................................(4)
dividing (3) by (4) we get
ap/aq = (q-p)/ (r-p)
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