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Q.
If the $m^{th}, n^{th}$ and $p^{th}$ terms of $A.P. \& G.P$. be equal & are respectively $x, y, z$, then which of the following hold good?
Sequences and Series
Solution:
Let first term of $A.P$. be $a$, with common difference $d$
$\therefore x = a +\left(m -1\right) d = t_{m}$
$y = a +\left(n -1\right) d = t_{n}$
$z = a +\left(p -1\right) d = t_{p}$
Further say, $A$ be the first term of $GP$. with common ratio $R$.
$\therefore x =AR^{m -1} = t_{m}$
$y= AR^{n -1} = t_{n}$
$z= AR^{p -1} = t_{p}$
$\therefore x^{y -z}\cdot y^{z -x} \cdot z^{x- y} = A^{0} R^{0} = 1$
$\therefore x^{y} y^{z} z^{x} = x^{z} y^{x} z^{y}$