Find the common ratio

Working with geometric sequences almost always comes down to two unknowns: the first term (a) and the common ratio (r). In this problem we use a pair of given terms to set up equations, then eliminate (a) by division — a clean technique that appears regularly in HSC Extension 1 questions. By the end you’ll see how quickly we can extract (r) and then backtrack to find (a_1).

Problem Statement

In a geometric sequence, (a_2=12) and (a_5=324). Find the common ratio (r) and the first term (a_1).

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Solutions

Recall the general term of a geometric sequence: (a_n = a r^{,n-1}), where (a) is the first term ((a_1)). Substituting the known terms gives us two equations.

From (a_n=ar^{n-1}),

$$a_2=ar=12,\qquad a_5=ar^4=324.$$

We want to find (r) without knowing (a) yet, so we divide the (a_5) equation by the (a_2) equation to cancel (a).

Divide to remove (a):

$$\frac{a_5}{a_2}=r^3=\frac{324}{12}=27 \implies r=3.$$

Now that we have the common ratio, we can substitute back into either original equation to find the first term. Using (a_2 = a r):

Then (a_1=a=\frac{12}{3}=4).

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Takeaways

• When a geometric sequence gives two terms, relate them through (a r^{n-1}) and divide the expressions to eliminate the first term.
• Solving (r^k = \text{number}) often reduces to taking a simple cube root (or square root) — always check for integer possibilities first in HSC problems.
• Once (r) is known, substitute back to find (a_1) using the lightest equation available.

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Further Readings

HSC Distributions, HSC Integrals, HSC Trigonometry, HSC Differential Equations