MathematicsChapter 4: Sequences and Series

Arithmetic Means

One arithmetic mean between a and b is A = (a+b)/2 (ordinary average).

Learning objectives

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

Definition

One arithmetic mean between a and b is A = a+b2 (ordinary average). To insert k arithmetic means between a and b, you make an AP of k+2 terms with first a and last b. Then d = b-ak+1. The means are a+d, a+2d, ..., a+kd. Meaning: you pad equal steps between two end numbers. The sum of those k means is k times the single AM (school identity).

Mathematical form

k=2, d=15-32+1=4

A = a+b2 (ordinary average). To insert k

d = b-ak+1. The means are a+d

d=15-32+1

Where / symbols

ARITHMETIC MEANS: One arithmetic mean between a and b is A = (a+b)/2 (ordinary average). To insert k arithmetic means between a and b, you make an AP of k+2 terms with first a and last b. Then d = (b-a)/(k+1). The means are a+d, a+2d, ..., a+kd. Meaning: you pad equal steps between two end numbers. The sum of those k means is k times the single AM (school identity).
1. ONE AM: Between 4 and 10: 7. Between -3 and 5: 1. The three terms 4, 7, 10 are an AP.
2. k MEANS: Total terms n = k+2. a_n = b = a+(n-1)d so d=(b-a)/(k+1). List them. Sum of k means = k (a+b)/2.

Relationship

one AM = a+b2
k means: d = b-ak+1

Overview

One mean, then many:

One Am

Between 4 and 10: 7. Between -3 and 5: 1. The three terms 4, 7, 10 are an AP.

Examples (a+b)/2 · Ends stay a and b · Middle is the mean

k MEANS

Total terms n = k+2. a_n = b = a+(n-1)d so d=b-ak+1. List them. Sum of k means = k a+b2.

Examples 3 means between 2 and 14: d=3 · 2,5,8,11,14 · Sum of means = 3*8=24

WHAT — ARITHMETIC MEANS

RULE — 1. ONE AM

Between 4 and 10: 7. Between -3 and 5: 1. The three terms 4, 7, 10 are an AP.

METHOD — 2. k MEANS

Total terms n = k+2. a_n = b = a+(n-1)d so d=b-ak+1. List them. Sum of k means = k a+b2.

WORKED EXAMPLE

Worked: two AMs between 3 and 15. k=2, d=15-32+1=4 means: 3+4=7 and 3+8=11 sequence: 3, 7, 11, 15

How to answer a arithmetic means 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

two AMs between 3 and 15.
k=2, d=15-32+1=4
means: 3+4=7 and 3+8=11
sequence: 3, 7, 11, 15

In simple words

k means need k+1 equal steps, not k steps. Divide by k+1.

Key points

one AM = a+b2
k means: d = b-ak+1
average
k+1 jumps from a to b

Exam preparation

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