Sequence And Series Question 193
Question: If $ p^{th},\ q^{th},\ r^{th} $ and $ s^{th} $ terms of an A.P. be in G.P., then $ (p-q),\ (q-r),\ (r-s) $ will be in
[MP PET 1993]
Options:
A) G.P.
B) A.P.
C) H.P.
D) None of these
Show Answer
Answer:
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.
BETA
