ARCH-MM-03: Coupled Arithmetic & Geometric Progressions and Sum to Infinity (S_∞)

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-MM-03: Why the 160s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~160s (Blows 90s Clock)

The common ratio is

r = \frac{u_2}{u_1} = \frac{8}{12} = \frac{2}{3}

. Since

|r| < 1

, substituting into

S_\infty = \frac{a}{1 - r}

gives

\frac{12}{1 - (\frac{2}{3})} = 36

(~95s).

The Fast ShortcutTarget: 45s (Saves 115s)

Line 1 (Ratio Difference Identity):

S_\infty = \frac{u_1^2}{u_1 - u_2}

directly without fractional

r

.
Line 2 (Cancel Factors):

\frac{12^2}{12 - (8)} = 36

(25s).

Where Students Lose Time: In 37 past UAT-UK questions (9.02% of Mathematics 2), candidates lose 115s by failing to reject divergent common ratios in s_infinity = a / (1 - r). Instead, apply MM2 with Maths 1 skill M4.10.

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