Geometric Progressions

A geometric progression multiplies by a constant ratio $r$ each step: $a$, $ar$, $ar^2$, … Test by dividing consecutive terms, constant quotient, GP confirmed.

The formulas

• nth term: $u_n = ar^{n-1}$
• Sum: $S_n = \dfrac{a(1 - r^n)}{1 - r}$ ($r \ne 1$), the twin form $\dfrac{a(r^n - 1)}{r - 1}$ is identical and tidier when $r > 1$
• Sum to infinity: $S_\infty = \dfrac{a}{1 - r}$, only for $|r| < 1$

Finding $r$: divide, then mind the $\pm$

A GP has $u_2 = 6$ and $u_5 = 48$. Find $r$ and $a$. $\dfrac{u_5}{u_2} = r^3 = 8 \to$ $r = 2$; $a = \dfrac{6}{r} =$ $3$

Dividing terms cancels $a$, the standard opening move. The subtlety: an even power gives two roots. $r^2 = 9 \to r = \pm 3$, and both must be considered unless the question excludes one (“all terms are positive” kills the negative; a sum-to-infinity requirement forces $|r| < 1$). Checking and stating the rejection is a mark; silently taking $+3$ is a gamble the mark scheme doesn’t reward.

Exponent discipline mirrors the AP’s off-by-one: $u_5 = ar^4$, not $ar^5$.

Growth and decay costumes

GPs arrive as compound interest ($r = 1.05$ for 5% growth), depreciation ($r = 0.8$ for 20% annual loss), bouncing balls (each bounce a fraction of the last). Translate the percentage into $r$ first, in writing. “value multiplies by 0.8 each year, so $r = 0.8$”, then the formulas take over. “When does the value first fall below RM5,000?” → $ar^{n-1} < 5000$, solved with logarithms: the GP topic’s secret exit into logs, and a favourite Paper 2 crossover.

AP–GP hybrids

“The 1st, 3rd and 7th terms of an AP form a GP”, write the three AP terms ($a$, $a + 2d$, $a + 6d$), impose the GP condition ($\text{middle}^2 = \text{product of outers}$, or equal ratios), and solve the resulting quadratic. The setup equation carries most of the marks; expect one solution to be degenerate ($d = 0$) and to be rejected with a reason.

Common mistakes

• The $\pm$ lost on even-power roots of $r$
• Exponent off-by-one ($ar^n$ instead of $ar^{n-1}$)
• Percentage changes translated to $r$ wrongly (5% growth is $r = 1.05$, not $0.05$)
• $S_n$ formula applied with $r = 1$ (it divides by zero, an $r = 1$ “GP” is constant; sum $= na$)
• Hybrid questions started without writing the AP terms first

Full topic context: Series notes.