Arithmetic and geometric sequences

TL;DR:

• An arithmetic sequence adds a common difference d. A geometric sequence multiplies by a common ratio r.

• Arithmetic: aₙ = a₁ + (n − 1)d and Sₙ = n(a₁ + aₙ)/2.

• Geometric: aₙ = a₁rⁿ⁻¹ and Sₙ = a₁(1 − rⁿ)/(1 − r).

A story goes that a teacher asked a young pupil to add the whole numbers from 1 to 100, hoping to keep the class busy. The pupil, who later became the mathematician Gauss, saw a shortcut and answered in moments. The shortcut is worth knowing, because it is the idea behind every formula for an arithmetic sum.

Two ways a list can grow

A sequence is a list of numbers in order. If each term is found by adding the same number, it is arithmetic: 5, 8, 11, 14, … adds 3 each time, so the common difference is d = 3. If each term is found by multiplying by the same number, it is geometric: 3, 6, 12, 24, … doubles each time, so the common ratio is r = 2.

To tell which one you have, check the gaps. Constant subtraction between neighbours means arithmetic. Constant division means geometric. If neither is constant, it is neither.

Worked example: the sum 1 + 2 + … + 100 = 5050

1. Write the sum forwards: 1 + 2 + 3 + … + 98 + 99 + 100.

2. Write it backwards underneath: 100 + 99 + 98 + … + 3 + 2 + 1.

3. Add the columns. Each pair sums to 101: 1 + 100, 2 + 99, 3 + 98, and so on.

4. There are 100 columns, so the two rows together total 100 × 101 = 10,100.

5. That counts the sum twice, so the sum is 10,100 / 2 = 5,050.

This gives the general rule for an arithmetic sum: Sₙ = n(a₁ + aₙ) / 2. With n = 100, a₁ = 1, aₙ = 100, we get 100 × 101 / 2 = 5050. A check on a smaller case: for 1 + 2 + 3 + 4 the formula gives 4 × 5 / 2 = 10, and indeed 1 + 2 + 3 + 4 = 10.

The nth term of an arithmetic sequence

The first term is a₁. The second is a₁ + d. The third is a₁ + 2d. The pattern is aₙ = a₁ + (n − 1)d, because you add d only n − 1 times to reach the nth term. For 5, 8, 11, … the 20th term is 5 + 19 × 3 = 62. The sum of the first 20 is 20 × (5 + 62) / 2 = 20 × 67 / 2 = 670.

Geometric sequences: terms and sums

The nth term of a geometric sequence is aₙ = a₁ rⁿ⁻¹. For 3, 6, 12, 24, … the 5th term is 3 × 2⁴ = 48. To add the first n terms, use Sₙ = a₁(1 − rⁿ)/(1 − r). For five terms: 3 × (1 − 32)/(1 − 2) = 3 × (−31)/(−1) = 93. Check by adding: 3 + 6 + 12 + 24 + 48 = 93.

Where does the formula come from? Multiply the sum by r and subtract: Sₙ − rSₙ = a₁ − a₁rⁿ, since all the middle terms cancel. Divide by 1 − r and you have it. It is the pairing trick again, in a different form.

Try one yourself: a countdown sequence

A sequence runs 100, 94, 88, … so a₁ = 100 and d = −6. When does it first go negative? Solve 100 − 6(n − 1) < 0: 6(n − 1) > 100, so n − 1 > 16.67 and n = 18. Check: a₁₈ = 100 − 6 × 17 = −2, while a₁₇ = 100 − 6 × 16 = 4. The sum of the 17 positive terms is 17 × (100 + 4) / 2 = 884. Quick test: the average term is 52, the middle one, and 17 × 52 = 884.

When the sum goes on forever

If |r| < 1, the powers rⁿ shrink toward zero, and the infinite sum becomes a₁ / (1 − r). For 1 + 1/2 + 1/4 + …, that is 1 / (1 − 1/2) = 2. Check with partial sums: 1, 1.5, 1.75, 1.875, 1.9375. They creep toward 2. If |r| ≥ 1, the sum does not settle.

How to check your answer

• Small case. Test your formula on the first three or four terms by adding by hand.

• Sign of d or r. A decreasing arithmetic sequence has negative d. A geometric sequence with negative r alternates signs.

• Count. The number of terms from 5 to 62 in steps of 3 is (62 − 5)/3 + 1 = 20.

• Size. A geometric sum with r above 1 should be dominated by its last terms.

Common mistakes

• Using n instead of n − 1 in the nth-term formula.

• Mixing up d and r, for example adding when the pattern multiplies.

• Using the arithmetic sum formula on a geometric sequence.

• Forgetting to divide by 2 in n(a₁ + aₙ)/2.

• Taking the sum of an infinite geometric series with |r| ≥ 1.

Where you will use it next

Sequences describe repeated growth. Adding a fixed amount is linear growth, and multiplying by a fixed ratio is exponential growth, as in exponential growth vs linear growth. Patterns that repeat in a cycle are the subject of sine and cosine graphs. Sums of many small terms are also the idea behind the integral in calculus.

Frequently asked questions

Is a constant sequence like 4, 4, 4 arithmetic or geometric?

Both. It has a common difference of 0 and a common ratio of 1.

Can a sequence be both arithmetic and growing exponentially?

No. Apart from constant sequences, an arithmetic sequence grows by a fixed amount, which is linear, and a geometric sequence with a ratio different from 1 grows or shrinks by a fixed factor.