01 Key Concepts
Series vs. Sequence
A sequence is a list of terms; a series is the SUM of those terms.
Arithmetic Series Sum Formula
S(n) = n/2 * (a(1) + a(n)), the number of terms times the average of the first and last term.
Geometric Series Sum Formula (finite)
S(n) = a(1) * (1 - r^n) / (1 - r), for r not equal to 1.
Infinite Geometric Series
If |r| < 1, an infinite geometric series converges to a finite sum: S = a(1) / (1 - r). If |r| >= 1, it does not converge.
Sigma Notation
A compact way to write a sum, such as 'the sum from i=1 to n of a(i)', representing a(1)+a(2)+...+a(n).
02 Key Formulas
- Arithmetic series: S(n) = n/2 * (a(1) + a(n))
- Geometric series (finite): S(n) = a(1)*(1-r^n)/(1-r)
- Geometric series (infinite, |r|<1): S = a(1)/(1-r)
03 Solved Examples
Example 1 Find the sum of the first 10 terms of the arithmetic sequence 3, 7, 11, 15, ...
- First find a(10): a(10) = 3 + (10-1)*4 = 39 (using the sequence formula).
- Apply the series formula: S(10) = 10/2 * (3 + 39) = 5 * 42.
Answer: S(10) = 210
Example 2 Find the sum of the first 5 terms of the geometric sequence 2, 6, 18, 54, ...
- Identify a(1)=2, r=3, n=5.
- S(5) = 2*(1-3^5)/(1-3) = 2*(1-243)/(-2) = 2*(-242)/(-2).
- = -484/-2.
Answer: S(5) = 242
Example 3 Find the sum of the infinite geometric series 8 + 4 + 2 + 1 + ...
- Identify a(1)=8, r=1/2 (since |r|<1, this converges).
- S = a(1)/(1-r) = 8/(1-0.5) = 8/0.5.
Answer: S = 16
04 Practice Questions
1Find the sum of the first 6 terms of 2, 4, 6, 8, ... (arithmetic).
42 (S=6/2*(2+12))
2Find the sum of the first 4 terms of 1, 2, 4, 8, ... (geometric).
15
3Find the sum of the infinite series 4 + 2 + 1 + 0.5 + ...
8
4What is required for an infinite geometric series to converge?
The absolute value of the common ratio must be less than 1
5Find the sum of the first 5 terms of 5, 10, 15, 20, ... (arithmetic).
75
📄 Series — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.