What is a Sequence and a Series?
A sequence is an ordered list of numbers following a pattern, while a series is the sum of those numbers. Arithmetic and geometric sequences are the two most common types you'll meet in algebra and calculus.
A sequence is an ordered list of terms (a₁, a₂, a₃, …); a series is the sum of a sequence's terms. In an arithmetic sequence terms differ by a constant d; in a geometric sequence terms are multiplied by a constant ratio r.
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Step-by-step worked examples
Find the sum of the first 10 terms of the arithmetic sequence 3, 7, 11, 15, …
a1 = 3, d = 4, n = 10 S_n = n/2·(2a1+(n-1)d) = 10/2·(2·3+9·4) = 5·(6+36) = 5·42 = 210
What is the 8th term of the sequence 5, 8, 11, 14, …?
a1=5, d=3 a_n = a1+(n-1)d a_8 = 5+7·3 = 5+21 = 26
Find the sum of the first 5 terms of the geometric sequence 2, 6, 18, 54, …
a1=2, r=3, n=5 S_n = a1·(r^n−1)/(r−1) = 2·(3^5−1)/(3−1) = 2·242/2 = 242
Flashcards
Quick quiz
Q1.What is the 6th term of 4, 7, 10, 13, …?
Q2.Sum of first 4 terms of 1, 2, 4, 8, …?
Q3.In an arithmetic sequence, what stays constant between terms?
Q4.A series is…
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Common mistakes
Sequence and series mean the same thing. — Correct: A sequence is a list of terms; a series is their sum.
Assuming every sequence is arithmetic or geometric. — Correct: Many sequences follow other rules (e.g. Fibonacci); check the pattern first.
Using r=1 in the geometric sum formula S_n=a1(r^n-1)/(r-1). — Correct: When r=1 the formula divides by zero; use S_n = n·a1 instead.
Starting term count from n=0 instead of n=1. — Correct: Standard convention: a1 is the first term, so a_n = a1+(n-1)d.
FAQ
What is a sequence in math?
A sequence is an ordered list of numbers that follow a specific rule, such as 2, 4, 6, 8, … where each term increases by 2.
What is the sequences and series formula?
The nth term of an arithmetic sequence is a_n = a1+(n−1)d, and the sum of its first n terms is S_n = n/2·(2a1+(n−1)d).
What are examples of sequences and series?
Examples include arithmetic sequences like 3, 7, 11, 15 and geometric sequences like 2, 6, 18, 54, along with their running sums (series).
How to calculate sequences and series?
Identify the pattern (arithmetic or geometric), find a1 and d or r, then apply the nth-term or sum formula to get any term or total.