Linear combinations of two GPs

We explore a classic link between sequences: linear combinations of two geometric progressions (GPs) produce terms that satisfy a second-order homogeneous recurrence. This connection sits squarely within the sequences and series topic of the HSC Extension 1 syllabus and provides a direct path to solving recurrence equations—by expressing unknown sequences as combinations of GPs with the characteristic roots. Working through the problem will sharpen your ability to move between closed-form expressions and recurrences, a skill that underpins many advanced problems in Extension 2 as well.

Problem Statement

Let $T_n=ar^{n-1}$ and $U_n=AR^{n-1}$. Define $W_n=T_n+U_n$.

• Show $(W_n)$ satisfies

$$W_{n+2}-(r+R)W_{n+1}+rRW_n=0.$$

• Solve

$$X_{n+2}-5X_{n+1}+6X_n=0,\quad X_1=5,\ X_2=13.$$

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Hints

For (ii), the characteristic roots are $2$ and $3$.

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Solutions

For (i), we simply substitute the definition of $W_n$ into the left-hand side and see that it collapses to zero. Because $W_n$ is the sum of two GPs with common ratios $r$ and $R$, the recurrence relation naturally emerges:

Substitute $W_n=ar^{n-1}+AR^{n-1}$ directly to verify (i).

For (ii), we recognise that the recurrence matches the form from part (i) with $r=2$ and $R=3$ (since $5=2+3$ and $6=2\times3$). Therefore its solution is a linear combination of the two GPs with those common ratios. We let

$$X_n=\alpha 2^{n-1}+\beta 3^{n-1}.$$

Using the given initial terms $X_1=5$ and $X_2=13$ gives the system

$$\alpha+\beta=5,\quad 2\alpha+3\beta=13$$

gives $\alpha=2,\ \beta=3$. Hence

$$X_n=2\cdot2^{n-1}+3\cdot3^{n-1}=2^n+3^n.$$

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Takeaways

• A sum of two geometric progressions with distinct ratios $r$ and $R$ satisfies $W_{n+2}-(r+R)W_{n+1}+rRW_n=0$, a second-order linear recurrence whose characteristic equation has roots $r$ and $R$.
• Conversely, a recurrence of the form $X_{n+2}-(r+R)X_{n+1}+rRX_n=0$ can be solved by writing $X_n=\alpha r^{n-1}+\beta R^{n-1}$ (provided $r\neq R$); the initial values determine $\alpha$ and $\beta$.
• This technique—expressing a sequence as a linear combination of geometric progressions—is a powerful algebraic shortcut that avoids more cumbersome iterative methods.

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Further Readings

HSC Mechanics, HSC Integrals, HSC Induction, HSC Functions