Geometric progressions · 4 marks

The sum of the first three terms of a GP is 1312 and their product is −1. Find the common ratio and the terms.

Answer: r = −34 with terms 43, −1, 34, or r = −43 with terms 34, −1, 43.

Step-by-step solution

Idea: Take the three terms as ar, a, ar. Their product is a³, which gives a at once; the sum then gives an equation for r.

Let the terms be ar, a, ar. Product = ar × a × ar = a³ = −1 ⇒ a = −1.1 mark
Sum: −1r − 1 − r = 1312 ⇒ 1r + r = −2512. Multiply by 12r: 12 + 12r² = −25r ⇒ 12r² + 25r + 12 = 0.1 mark
Split the middle term (16 × 9 = 144 = 12 × 12): 12r² + 16r + 9r + 12 = 4r(3r + 4) + 3(3r + 4) = (4r + 3)(3r + 4) = 0 ⇒ r = −34 or −43.1 mark
r = −34: terms −1−3/4 = 43, −1, 34. r = −43: terms 34, −1, 43 (the same numbers in reverse order).1 mark

Common ratio −3/4 with terms 4/3, −1, 3/4; or common ratio −4/3 with terms 3/4, −1, 4/3.

Check: 43 − 1 + 34 = 16 − 12 + 912 = 1312 ✓; 43 × (−1) × 34 = −1 ✓.

Let the terms be ar, a, ar

Product: a³ = −1 ⇒ a = −1

Sum: −1r − 1 − r = 1312 ⇒ 12r² + 25r + 12 = 0

(4r + 3)(3r + 4) = 0 ⇒ r = −34 or −43

∴ r = −34: terms 43, −1, 34; r = −43: terms 34, −1, 43

Taking the terms as a, ar, ar²: it works but gives a harder equation. ar, a, ar makes the product simply a³.
Taking a = 1 from a³ = −1. The cube root of −1 is −1.
Keeping only one value of r. Both −34 and −43 satisfy the conditions.

MCQ (1 mark)

Three terms of a GP have product 64. The middle term is

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(B) 4
ar × a × ar = a³ = 64 ⇒ a = 4.

The product of three consecutive terms of a GP is 1000 and their sum is 35. Find the terms.

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a³ = 1000 ⇒ a = 10. 10r + 10 + 10r = 35 ⇒ 2r² − 5r + 2 = 0 ⇒ r = 2 or 12. Terms: 5, 10, 20 (or 20, 10, 5).