Sequences and Series

Arithmetic sequences

In an arithmetic sequence you add the same number every time. That number is the common difference, d. Find it by subtracting any term from the next one. In 4, 7, 10, 13, and so on, d = 7 − 4 = 3. To reach term n from the first term you take n − 1 steps, so the nth term is a_n = a₁ + (n − 1)d.

Geometric sequences

In a geometric sequence you multiply by the same number every time. That number is the common ratio, r. Find it by dividing any term by the one before it. In 3, 6, 12, 24, and so on, r = 6/3 = 2. After n − 1 multiplications the nth term is a_n = a₁ · r^{n − 1}.

To tell the two kinds apart, test both: if the differences match, it is arithmetic; if the ratios match, it is geometric. The sequence 1, 4, 9, 16, and so on is neither.

Explicit and recursive formulas

An explicit formula gives any term straight from its position n, like a_n = 3n + 1. A recursive formula gives the first term and a rule for getting each term from the one before, like a₁ = 4 and a_n = a_{n − 1} + 3. Both describe the same sequence 4, 7, 10, 13, and so on. The recursive form is easy to follow step by step; the explicit form is faster for a far-off term like a₁₀₀.

Arithmetic series

To add the terms of an arithmetic sequence, pair the first term with the last, the second with the second-to-last, and so on: every pair has the same total. That gives the sum of the first n terms, S_n = n(a₁ + a_n)/2: the number of terms times the average of the first and last term.

Finite geometric series

For a geometric sequence with ratio r ≠ 1, the sum of the first n terms is S_n = a₁(1 − rⁿ)/1 − r. Why it works: write the sum S, multiply it by r, and subtract. Almost every term cancels, leaving S − rS = a₁ − a₁ · rⁿ.

Sigma notation

The Greek capital letter sigma, Σ, means “add up”. In 5Σk = 1(2k + 1), the letter k is the index: put k = 1, 2, 3, 4 and 5 into 2k + 1 and add the results. The number below Σ is where k starts and the number above is where it stops, so the number of terms is the top minus the bottom plus 1.

Growth and decay

A geometric sequence is exponential growth or decay counted in whole steps. The formula a_n = a₁ · r^{n − 1} has the same shape as the exponential function y = a · b^x from Functions and Their Graphs: the ratio r plays the part of the base b. If r > 1 the terms grow; if r is between 0 and 1 they decay. In the same way an arithmetic sequence is a linear function with slope d, which is why its dots sit on a line.

Common mistakes

Practice problems

Printable worksheet (PDF)

Try each one, then open the answer to check your work.

1. Find the common difference and a₃₀ for the sequence 6, 11, 16, 21, and so on.
   Show answer

   d = 511 − 6 = 5a₃₀ = 6 + 29 · 5 = 151
2. Find the common ratio and a₆ for the sequence 5, −10, 20, −40, and so on.
   Show answer

   r = −2−10/5 = −2a₆ = 5 · (−2)⁵ = −160
3. Is 2, 6, 18, 54, and so on arithmetic or geometric? Write its explicit formula.
   Show answer

   r = 3The ratios are all 3, so it is geometrica_n = 2 · 3^{n − 1}
4. Write an explicit formula for 20, 17, 14, 11, and so on.
   Show answer

   a_n = 20 + (n − 1)(−3)d = −3a_n = −3n + 23Any equal form is fine
5. A sequence has a₁ = 10 and a_n = a_{n − 1} − 4. List the first four terms.
   Show answer

   a₂ = 10 − 4 = 6Subtract 4 each timea₃ = 6 − 4 = 2a₄ = 2 − 4 = −2The terms are 10, 6, 2, −2
6. Find the sum of the first 25 terms of 2, 5, 8, 11, and so on.
   Show answer

   a₂₅ = 2 + 24 · 3 = 74Last termS₂₅ = 25(2 + 74)/2 = 950
7. Find the sum of the first 5 terms of 4, 12, 36, and so on.
   Show answer

   S₅ = 4(1 − 3⁵)/1 − 3r = 3S₅ = 4(−242)/−2 = 484
8. Find the value of 8Σk = 1(3k − 2).
   Show answer

   3(8) − 2 = 228 terms: k = 1 gives 1 and k = 8 gives 228(1 + 22)/2 = 92
9. A town of 1200 people grows by 5% a year. Write a formula for the population at the start of year n.
   Show answer

   a_n = 1200 · 1.05^{n − 1}Growing 5% means multiplying by r = 1.05

History: adding long lists

A famous story about Carl Friedrich Gauss says that as a young schoolboy he added the whole numbers from 1 to 100 almost at once by noticing 50 pairs that each add to 101. The MacTutor History of Mathematics archive tells it this way; the details of such childhood stories are hard to confirm, so treat it as a story, but the pairing idea is exactly the arithmetic series formula.

The sigma sign has a better-recorded start. According to Jeff Miller's notes on the earliest uses of mathematical symbols, Leonhard Euler used Σ for a sum in his 1755 book Institutiones calculi differentialis, choosing it to match the Greek letter Δ he used for differences.

Connections

Arithmetic sequences are linear functions with whole-number inputs, from Graphing Linear Equations: the common difference is the slope. Geometric sequences are exponential functions, from Functions and Their Graphs, and their powers follow the rules in Exponents and Radicals.

Solving for n, like finding which term equals 100, uses Linear Equations. The geometric series formula is one fraction, as in Rational Expressions and Equations.

Sources

• J J O'Connor and E F Robertson, “Johann Carl Friedrich Gauss”, MacTutor History of Mathematics, University of St Andrews: Gauss biography (the 1 to 100 story)
• Jeff Miller, “Earliest Uses of Symbols of Operation”, hosted by MacTutor: Symbols of operation (Euler and Σ, 1755)
• Common Core State Standards for Mathematics: Interpreting Functions (F-IF.A.3), Building Functions (F-BF.A.2), Linear, Quadratic, and Exponential Models (F-LE.A.2), Seeing Structure in Expressions (A-SSE.B.4)

Credits: all diagrams on this page are original SVG drawings made for Jugami. Every answer was double-checked.