Arithmetical
Progression

A
series of numbers is in arithmetical progression if each number in the series
is a constant increase or decrease over its preceding number. Put another way,
if we take any number in the series and the number that follows it in the
series and find the difference, the series is an arithmetical progression if
the differences are constant. For example, Macdonald gives the series 2, 5, 8,
11, 14, 17, 20, 23, 26, 29, 32 and 35 where the difference between any number
and the succeeding number is 3.

Of
interest to Macdonald, or at least to his teacher, is to find the sum:

For
the first time in the notebook, Macdonald uses algebra. He, or his teacher,
expresses this foray into algebra quite quaintly:

“It
will be of the greatest advantage in the extension of arithmetic to apply the
use of letters to denote certain quantities until they are determined, that we
may express our ideas clearly, fully and briefly by applying to them the signs
of arithmetic.”

The
letters Macdonald uses are:

first term

difference between
succeeding terms

number of terms

sum of the terms

In
general

In
the numerical example, , , and .

The
usual way to obtain a compact form of the sum is first to write the sum in
reverse order

Summing
the two together yields

so that

This
is not quite the route that Macdonald took. His derivation is slightly
different. First he assumes that is an even number. Then he adds the term and the term, the term and the term, the term and the term, and so on. In each case the sum is , a constant. Since
there are pairs of terms then the formula for is obtained.

When
is odd, Macdonald explains how to find the sum
by using a specific numerical example, and then says the method can be
generalized to get the same formula for . Essentially, when is odd there are pairs ( and , term and , and so on) that
sum to . This leaves the
middle term which is the average of the first and
last terms, or . Adding them all
together yields

the same formula as before.

In
the sum there are four terms (, , , and ). Macdonald
addresses the problem of given values for any three of these terms, solve for
the fourth. With one exception the solution is achieved by a straightforward
manipulation of the terms in the equation for . The exception is
the solution for given , and , which is the
solution to a quadratic equation. In this case the equation can be written as

To
solve for Macdonald completes the square and obtains the
equation

which can be rearranged to obtain

This yields

The
consideration of this particular topic is odd in that Macdonald does not cover
the concept of square roots until later in the
notebook. When he does cover square roots, arithmetical progressions are not
used in any numerical examples.