Geometric progressions

In a geometric progression (G.P.) each term is the one before it multiplied by a fixed number r, the common ratio. An A.P. adds the same number each time; a G.P. multiplies by the same number. No term of a G.P. is zero.

• 3, 6, 12, 24, ... has a = 3 and r = 2.
• To test a list, divide each term by the one before it: 1/9, –1/3, 1, –3 gives (–1/3) ÷ (1/9) = –3 every time, so it is a G.P. with r = –3. A negative ratio is fine; it just makes the signs take turns.

The nth term. The terms are a, ar, ar², ar³, ...: the first term has no r, the second has r¹, the third r². So the power of r is always one less than the position:

• aₙ = arⁿ⁻¹.
• 8th term of 3, 6, 12, ...: 3 × 2⁷ = 384.
• Which term of 3, 6, 12, ... is 768? 3 × 2ⁿ⁻¹ = 768, so 2ⁿ⁻¹ = 256 = 2⁸, n – 1 = 8, and it is the 9th term. Write both sides as powers of the same base and compare the powers.

Two terms given. If the 2nd term is 6 and the 5th is 162, then ar = 6 and ar⁴ = 162. Divide the second by the first: a cancels and r³ = 27, so r = 3 and a = 2.

Sum of n terms. For r ≠ 1:

• Sₙ = a(rⁿ – 1)/(r – 1), easier when r > 1.
• Sₙ = a(1 – rⁿ)/(1 – r), the same formula, easier when r < 1.
• If r = 1 every term is a, so Sₙ = na.

Examples:

• 2 + 6 + 18 + ... to 6 terms: 2(3⁶ – 1)/(3 – 1) = 3⁶ – 1 = 728.
• 81 + 27 + 9 + 3 + 1: a = 81, r = 1/3, so S₅ = 81(1 – 1/243)/(2/3) = 121. Adding by hand gives the same.
• How many terms of 3 + 6 + 12 + ... make 381? 3(2ⁿ – 1) = 381, so 2ⁿ = 128 = 2⁷ and n = 7.

Three terms with a known product: take a/r, a, ar. When multiplied, the r's cancel and a comes straight out. Three numbers in G.P. have product 216 and sum 19:

• (a/r) × a × ar = a³ = 216, so a = 6.
• 6/r + 6 + 6r = 19 gives 6r² – 13r + 6 = 0, so r = 3/2 or 2/3.
• Either way the numbers are 4, 6, 9.

Sums like 3 + 33 + 333 + ... are not a G.P., but can be turned into one. Take out 3 and multiply and divide by 9 to get nines: (3/9)(9 + 99 + 999 + ...). Then 9 = 10 – 1, 99 = 10² – 1, and so on, so the sum to n terms is (1/3)[(10 + 10² + ... + 10ⁿ) – n] = (1/3)[10(10ⁿ – 1)/9 – n]. Check with n = 2: (1/3)(110 – 2) = 36 = 3 + 33.