Central symmetry in a GP

The product of the terms of a geometric progression often reveals elegant symmetries that can turn a lengthy calculation into a single step. This problem, from the sequences and series topic in the HSC Extension 1 syllabus, shows you how to compactly express a product of any number of terms, prove that pairs of terms equidistant from the ends have the same product, and then exploit that symmetry when the number of terms is odd. Working through these ideas will sharpen your algebraic manipulation with index laws and develop the intuition to spot shortcuts when a GP has a middle term.

Problem Statement

In a GP, define $P_n=T_1T_2\cdots T_n$.

• Show $P_n=a^n r^{\frac{n(n-1)}2}$.
• Show $T_kT_{n-k+1}=T_1T_n$.
• If $n=11$ and middle term $T_6=5$, find $P_{11}$.

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Hints

Use $T_k=ar^{k-1}$ and pair symmetric terms.

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Solutions

We begin by writing the product $P_n$ directly from the general term $T_i = ar^{i-1}$ and collecting powers of $a$ and $r$ separately.

$$P_n=a(ar)\cdots(ar^{n-1})=a^n r^{0+1+\cdots+(n-1)}=a^n r^{\frac{n(n-1)}2}.$$

For the symmetry property, we take a pair of terms that are symmetric in the list – one $k$ steps from the start, the other $k$ steps from the end – and multiply their expressions.

$$T_kT_{n-k+1}=ar^{k-1}\cdot ar^{n-k}=a^2r^{n-1}=T_1T_n.$$

Finally, we use the fact that an odd number of terms creates a natural middle term: paired terms multiply to the same constant and the unpaired middle term can be raised to the total power to give the whole product.

For odd $n=11$, the product is the middle term raised to the power $n$:

$$P_{11}=T_6^{11}=5^{11}.$$

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Takeaways

• The product of the first $n$ terms of a GP can be written as $a^{\,n}r^{\frac{n(n-1)}2}$ by summing the exponents from a simple arithmetic series.
• In any GP, terms symmetric about the centre satisfy $T_k T_{n-k+1} = T_1 T_n$, which is independent of $k$.
• When $n$ is odd, the entire product equals the middle term raised to the $n$‑th power – a rapid shortcut once symmetry is established.

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Further Readings

HSC Differential Equations, HSC Trigonometry, HSC Last Resorts, HSC Polys Ext 1