MathematicsChapter 4: Sequences and Series

Sum of An Arithmetic Series

S_n = n/2 * (2a + (n-1)d) = n/2 * (first + last).

Learning objectives

State the main ideas and notation used in sum of an arithmetic series.
Apply the rules of sum of an arithmetic series in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

S_n = n/2 (2a + (n-1)d) = n/2 (first + last). Meaning: n copies of the average of the first and last term. Use the first form if you know a and d; use first+last if you know the ends. Gauss's story: 1+...+100 = 1002 * 101.

Mathematical form

a=2, d=3, n=20

S₂0 = 202 * (4 + 19*3)

S_n = n/2 * (2a + (n-1)d)

n=20 S₂0

Where / symbols

SUM OF AN ARITHMETIC SERIES: S_n = n/2 * (2a + (n-1)d) = n/2 * (first + last). Meaning: n copies of the average of the first and last term. Use the first form if you know a and d; use first+last if you know the ends. Gauss's story: 1+...+100 = 100/2 * 101.
1. WITH a AND d: Plug n. Keep /2 till you have an even number if you want integers. S_n is not a_n.
2. WITH ENDS: If last term l is given, S_n = n(a+l)/2. Find n first from a_n = l if needed. Word problems: seats in a theatre, logs in a stack (school).

Relationship

S_n = n/2 * (2a+(n-1)d)

Overview

Two formulas, same sum:

WITH a AND d

Plug n. Keep /2 till you have an even number if you want integers. Sn is not an.

Examples Sum of 10 terms · d negative: still OK · S_n grows like n^2

With Ends

If last term l is given, Sn = n(a+l)/2. Find n first from an = l if needed. Word problems: seats in a theatre, logs in a stack (school).

Examples 1 to n: n(n+1)/2 · Find n from last term · Check a_n

WHAT — SUM OF AN ARITHMETIC SERIES

RULE — 1. WITH a AND d

Plug n. Keep /2 till you have an even number if you want integers. Sn is not an.

METHOD — 2. WITH ENDS

If last term l is given, Sn = n(a+l)/2. Find n first from an = l if needed. Word problems: seats in a theatre, logs in a stack (school).

WORKED EXAMPLE

Worked: sum of first 20 terms of 2, 5, 8, ... a=2, d=3, n=20 S₂0 = 202 * (4 + 19*3) = 10 * (4+57) = 10*61 = 610

How to answer a sum of an arithmetic series question

Identify the given quantities, notation, and required result.
Choose the matching rule and write it before substitution.
Work one algebraic step at a time, keeping brackets and signs clear.
Check restrictions, simplify the result, and verify it where possible.

Worked example

sum of first 20 terms of 2, 5, 8, ...
a=2, d=3, n=20
S₂0 = 202 * (4 + 19*3)
= 10 * (4+57) = 10*61 = 610

In simple words

S_n uses n/2, not n. Forgetting /2 doubles the answer.

Key points

S_n = n/2 * (2a+(n-1)d)
also n/2 * (a1+a_n)
n/2 times the bracket
average of ends * n

Exam preparation

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