A) G.P.
B) A.P.
C) H.P.
D) None of these
Correct Answer: A
Solution :
If \[a\] and d be the first term and common difference of the A.P. Then \[{{T}_{p}}=a+(p-1)d,\ {{T}_{q}}=a+(q-1)d\] and \[{{T}_{r}}=a+(r-1)d\]. If \[{{T}_{p}},\ {{T}_{q}},\ {{T}_{r}}\] are in G.P. Then its common ratio \[R=\frac{{{T}_{q}}}{{{T}_{p}}}=\frac{{{T}_{r}}}{{{T}_{q}}}=\frac{{{T}_{q}}-{{T}_{r}}}{{{T}_{p}}-{{T}_{q}}}\] \[=\frac{[a+(q-1)d]-[a+(r-1)d]}{[a+(p-1)d]-[a+(q-1)d]}=\frac{q-r}{p-q}\] Similarly, we can show that \[R=\frac{q-r}{p-q}=\frac{r-s}{q-r}\] Hence \[(p-q),\ (q-r),\ (r-s)\] be in G.P.
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