A-Level Sequences and series: worked solution

Question

Let an arithmetic series have common difference d and first term a (both non-zero); a convergent geometric series has first term b > 0 and non-zero common ratio r. Let S be the geometric series' sum of n terms. Its 3rd, 6th and 10th terms equal the arithmetic series' 7th, 10th and 11th terms respectively. Show that r satisfies 3r^7 - 4r^3 + 1 = 0 and solve it to 4 decimal places.

Step 1: write down the terms being matched. The geometric series is b, br, br², br³, ... so its nth term is br^(n-1). Its 3rd, 6th and 10th terms are br², br⁵ and br⁹. The arithmetic series is a, a + d, a + 2d, ... so its nth term is a + (n-1)d. Its 7th, 10th and 11th terms are a + 6d, a + 9d and a + 10d. Step 2: match them in the order given, producing three equations. a + 6d = br² ... (1) a + 9d = br⁵ ... (2) a + 10d = br⁹ ... (3) Step 3: eliminate a by subtracting consecutive equations. (2) - (1): 3d = br⁵ - br² = br²(r³ - 1) (3) - (2): d = br⁹ - br⁵ = br⁵(r⁴ - 1) Step 4: eliminate d. The first line says 3d = br²(r³ - 1); the second says d = br⁵(r⁴ - 1), so 3d = 3br⁵(r⁴ - 1). Setting the two expressions for 3d equal: 3br⁵(r⁴ - 1) = br²(r³ - 1) Divide both sides by br². This is allowed because b > 0 and r ≠ 0: 3r³(r⁴ - 1) = r³ - 1 3r⁷ - 3r³ = r³ - 1 3r⁷ - 4r³ + 1 = 0 (as required) Step 5: solve the equation. r = 1 is an obvious root, since 3(1) - 4(1) + 1 = 0. It must be rejected: the geometric series converges only if |r| < 1, and r = 1 would also force d = 0, which is not allowed. Look for a root with |r| < 1 by testing for a sign change: at r = 0.66: 3r⁷ - 4r³ + 1 = +0.0137 at r = 0.67: 3r⁷ - 4r³ + 1 = -0.0212 The expression changes sign, so a root lies between 0.66 and 0.67. Narrowing it (by repeated bisection or a graphing calculator) gives r = 0.6639 to 4 decimal places. No negative root qualifies: for -1 < r < 0 the term -4r³ is positive and 3r⁷ is small, so the expression stays positive (for example at r = -0.5 it equals 1.477), giving no sign change there. So the only admissible root is r = 0.6639.