Q.If the $m$th and $n$th terms of a G.P are $n$ and $m$ respectively, show that its $(m+n)$th term is $\left(\dfrac{n^m}{m^n}\right)^{\frac{1}{m-n}}$.

Selection of Terms of a G.P. — First Principles

When you first meet a geometric progression, you see a sequence like $2, 4, 8, 16, 32$ — each term is the previous one multiplied by a fixed number (the common ratio $r$). The natural way to write a G.P. is:

where $a$ is the first term and $r$ is the common ratio. That works perfectly when you know exactly which term is the first one.

But what if you don't know where the sequence starts? What if you only know that three numbers are in G.P., but you don't know their positions? For example, suppose three numbers in G.P. have a product of $216$ and a sum of $26$. If you blindly call them $a, ar, ar^2$, you get:

That's neat — the middle term pops out immediately. But then the sum equation becomes $a + 6 + ar^2 = 26$, which is $a + ar^2 = 20$. Substituting $a = 6/r$ gives $6/r + 6r = 20$, a quadratic in $r$ that works fine. So the standard form is usable.

But there is a smarter way — a way that makes the algebra symmetrical and often avoids fractions entirely.

The Symmetrical Selection

When you are dealing with an odd number of terms in G.P., you can centre them around the middle term. Instead of writing three terms as $a, ar, ar^2$, write them as:

Here $a$ is the middle term, and $r$ is still the common ratio. The product of these three terms is:

That's even cleaner than before — the $r$ cancels completely. For the same problem (product $216$), you get $a^3 = 216$, so $a = 6$ instantly. Then the sum is:

Multiply through by $r$: $6 + 6r^2 = 20r$, or $3r^2 - 10r + 3 = 0$, giving $r = 3$ or $r = \frac{1}{3}$. The numbers are $2, 6, 18$ (or $18, 6, 2$).

Extending to More Terms

The same idea works for any odd number of terms. For five terms in G.P., write:

The product is $a^5$. The symmetry means that if you know the product, you get $a$ directly. If you know the sum, you get an equation in $r$ that is often simpler than the alternative.

For four terms (an even number), symmetry is less natural, but you can still choose a form that simplifies the product. A common choice is:

Here the common ratio between consecutive terms is $r^2$, and the product is $a^4$. This is useful when the product is given.

The Precise Statement

When you are asked to "select terms of a G.P." in a problem, you are choosing a representation that makes the given conditions easiest to apply. The general principle is: …