Arithmetic and geometric progressions

Progressions are sequences of numbers built by a fixed rule. In an arithmetic progression the same number is added each time; in a geometric progression we multiply by the same number. In exams knowing the formulas is not enough: you need to see relations between terms and set up the right system.

§1. Arithmetic progression and its general term

An arithmetic progression is a sequence in which every term from the second on is obtained by adding the same number d (the common difference) to the previous one: aₙ₊₁ = aₙ + d.

• d > 0 — increasing, d < 0 — decreasing, d = 0 — constant.

Going from a₁ to aₙ we add d exactly n − 1 times, hence:

Characteristic property: each term (except the first) is the arithmetic mean of its neighbours:

This is used to check whether three numbers form a progression: for a, b, c we need 2b = a + c.

Worked example

a₃ = 7, a₇ = 19. Find a₂₀.

1. a₇ − a₃ = 4d = 12 → d = 3.
2. a₁ = a₃ − 2d = 1.
3. a₂₀ = 1 + 19·3 = 58.

§2. Sum of an arithmetic progression

Gauss's trick: write the sum twice — forwards and backwards — and add. Every pair equals a₁ + aₙ, there are n pairs, and the result is twice the sum:

The reverse relation is also useful:

Worked example

Find the sum of all two-digit multiples of 7.

1. The first is 14, the last 98, d = 7.
2. Number of terms: (98 − 14)/7 + 1 = 13.
3. S = (14 + 98)·13/2 = 56·13 = 728.

Worked example

a₁₀ + a₁₂ = 25, a₂₀ + a₂₂ = 45. Find a₁₁ + a₂₁.

1. By the property: a₁₀ + a₁₂ = 2a₁₁ → a₁₁ = 12.5.
2. Similarly 2a₂₁ = 45 → a₂₁ = 22.5.
3. a₁₁ + a₂₁ = 35.

§3. Geometric progression

A geometric progression: each term is obtained by multiplying the previous one by the same number q ≠ 0 (the common ratio): bₙ₊₁ = bₙ·q, b₁ ≠ 0.

Where does the sum formula come from? Multiply Sₙ by q and subtract Sₙ: the middle terms cancel, qSₙ − Sₙ = b₁qⁿ − b₁.
If q < 0, the signs of the terms alternate — remember this in your answer.

Worked example

In a geometric progression b₃ = 18, S₃ = 26. Find b₁.

1. b₁ + b₁q = 8, b₁q² = 18.
2. Divide: (1 + q)/q² = 4/9 → 4q² − 9q − 9 = 0.
3. D = 81 + 144 = 225, q = (9 ± 15)/8 → q = 3 or q = −3/4.
4. b₁ = 18/q²: b₁ = 2 or b₁ = 32. Check: 32 − 24 + 18 = 26 — correct.

§4. Infinite decreasing geometric progression

If |q| < 1, then qⁿ → 0 and Sₙ tends to a finite number:

Application — converting a recurring decimal to a fraction:

• 0.(3) = 0.3 + 0.03 + … = 0.3/(1 − 0.1) = 1/3;
• 0.(27) = 0.27/(1 − 0.01) = 27/99 = 3/11.

Worked example

The sum of an infinite decreasing geometric progression is 12 and its first term is 4. Find the ratio.

1. 4/(1 − q) = 12 → 1 − q = 1/3.
2. q = 2/3 (the condition |q| < 1 holds).

§5. Word problems

Rule of thumb: "increases by the same amount each time" — arithmetic; "increases so many times / by a percentage" — geometric.

Worked example

The first row of an amphitheatre has 20 seats, and each next row has 2 more. How many seats are there in 15 rows?

1. a₁ = 20, d = 2, n = 15.
2. S = (2·20 + 14·2)·15/2 = 68·15/2 = 510.

§6. The characteristic property and mixed problems

Every term of a progression (except the first) can be expressed through its neighbours. This property describes the progression completely, so it is called the characteristic property:

‘Mixed’ problems combining both properties are common. Choosing good notation is half the battle:

• Three numbers in arithmetic progression are conveniently written as a − d, a, a + d — their sum is 3a, so d disappears at once.
• Three numbers in geometric progression are conveniently written as b/q, b, bq — their product is b³.
• If two terms are given (aₘ and aₖ), the common difference comes from their difference: aₖ − aₘ = (k − m)d.

A problem may have two answers: if a quadratic gives two values of d (or q), check both against the conditions and keep those that fit. If the problem does not say ‘increasing’ or ‘with positive terms’, both answers are valid.

Worked example

Three numbers with sum 15 form an arithmetic progression. If 1, 4 and 19 are added to them respectively, a geometric progression results. Find the numbers.

1. The numbers are 5 − d, 5, 5 + d (since 3a = 15 → a = 5).
2. The new numbers: 6 − d, 9, 24 + d.
3. Characteristic property: 9² = (6 − d)(24 + d) → 81 = 144 − 18d − d² → d² + 18d − 63 = 0.
4. d = 3 or d = −21.
5. d = 3: 2, 5, 8 → 3, 9, 27 (q = 3). d = −21: 26, 5, −16 → 27, 9, 3 (q = 1/3). Both answers are valid.

Check yourself

1. a₁ = 5, d = −3. Find a₁₂.

2. Find the sum of the first 20 terms of 2, 5, 8, ….

3. b₁ = 3, q = 2. Find b₆ and S₆.

4. 1, x, 9 is a geometric progression with positive terms. x = ?

5. Find the sum 8 + 4 + 2 + ….

6. In an arithmetic progression Sₙ = n² + 3n. Find a formula for aₙ.

7. Three numbers in arithmetic progression have sum 15 and product 80. Find them.

8. In an arithmetic progression a₃ = 7 and a₇ = 19. Find a₁ and d.

Common mistakes

• Writing aₙ = a₁ + nd — the correct factor is (n − 1)d.
• Counting terms as (last − first)/d and forgetting + 1.
• Losing the negative-q case (in the example above the second answer is b₁ = 32).
• Using S = b₁/(1 − q) when |q| ≥ 1.
• Taking a square root of bₙ² = bₙ₋₁·bₙ₊₁ without considering the sign (if positivity is not given, ±).
• Discarding the second value of d (or q) from a quadratic without checking — it may satisfy the conditions too.