Suppose \( a_{1}, a_{2}, a_{3} \) are in A.P. and \( b_{1}, b_{2}, b_{3} \) are in H.P. and let ...
Suppose \( a_{1}, a_{2}, a_{3} \) are in A.P. and \( b_{1}, b_{2}, b_{3} \) are in H.P. and let \( \Delta=\left|\begin{array}{lll}a_{1}-b_{1} & a_{1}-b_{2} & a_{1}-b_{3} \\ a_{2}-b_{1} & a_{2}-b_{2} & a_{2}-b_{3} \\ a_{3}-b_{1} & a_{3}-b_{2} & a_{3}-b_{3}\end{array}\right| \), then
(a) \( \Delta \) is independent of \( a_{1}, a_{2}, a_{3} \)
(b) \( a_{1}-\Delta, a_{2}-2 \Delta, a_{3}-3 \Delta \) are in H.P.
(c) \( b_{1}+\Delta, b_{2}+\Delta^{2}, b_{3}+\Delta \) are in A.P.
(d) \( \Delta \) is dependent of \( b_{1}, b_{2}, b_{3} \)
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