Consider the idea of inserting arithmetic means between two numbers.

MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series

Problem

Consider the idea of inserting arithmetic means between two numbers. (i) Define an arithmetic mean between two given numbers. (ii) Prove that if

k

arithmetic means are inserted between two distinct numbers

p

and

q

, then the entire set of

k+2

numbers forms an arithmetic progression. (iii) Three arithmetic means are to be inserted between

4

and

19

. Calculate the common difference of the resulting progression.