If $a$ and $b$ are two numbers,$A$ is the arithmetic mean,and $S$ is the sum of $n$ arithmetic means between $a$ and $b$,then what does $S/A$ depend on?

If $\frac{3 + 5 + 7 + \dots + (2n + 1)}{5 + 8 + 11 + \dots + (3n + 2)} = 7$,find the value of $n$. (Note: The original problem implies $n$ terms in the numerator and $10$ terms in the denominator as per the provided solution structure).

If $a, b, c$ are positive real numbers such that $ab^2c^3 = 64$,what is the minimum value of $(1/a + 2/b + 3/c)$?

Find the mean of the first $n$ terms of the $A$.$P$. $a, (a + d), (a + 2d), \dots$

In an $A.P.$,if the $m^{\text{th}}$ term is $n$ and the $n^{\text{th}}$ term is $m$,where $m \neq n$,find the $p^{\text{th}}$ term.

Which term of the sequence $(-8 + 18i), (-6 + 15i), (-4 + 12i), \dots$ is purely imaginary (in $^{th}$)?