Sequences and Mathematical Induction

Arithmetic Sequences

We say that the sequence

$$ a_1, a_2, \cdots, a_n, \cdots $$

is an arithmetic sequence (also called an arithmetic progression) if and only if there is a real number \(d\) such that

$$ a_{k + 1} - a_k= d $$

for every positive number \(k\). The number \(d\) is called the common difference.

Thus the general term of an arithmetic sequence is given by

$$ a_n = a_1 + (n - 1)d $$

where \(a_1\) is the first term, and \(d\) is the common difference.

Sum of Arithmetic Sequences

Consider an arithmetic sequence \(a_1, a_2, \cdots, a_n\) with a common difference \(d\). The sum of its elements is obtained as follows

$$ S_n = a_1 + (a_1 + d) + (a_1 + 2d) + \cdots + (a_n - 2d) + (a_n - d) + a_n $$

We write this sum in reverse:

$$ S_n = a_n + (a_n - d) + (a_n - 2d) + \cdots + (a_1 + 2d) + (a_1 + d) + a_1 $$

Add the two equations to produce

$$ 2S_n = (a_1 + a_n) + (a_1 + a_n) + (a_1 + a_n) + \cdots + \\ + (a_1 + a_n) + (a_1 + a_n) + (a_1 + a_n) $$

That is

$$ 2S_n = n(a_1 + a_n) $$

from which we obtain a sum formula:

$$ S_n = \frac{n(a_1 + a_n)}{2} $$

Geomemtric Sequences

We say the sequence \(a_1, a_2, \cdots, a_n\) is a geometric sequence (or geometric progression) if and only if there is a nonzero real number (r\ such that)

$$ a_{k + 1} = r a_k $$

for every positive integer \(k\). The nonzero real number \(r\) is called the common ratio of the sequence.

Thus the general term of a geometric sequence is given b

$$ a_n = a_1 r^{n - 1} $$

Sum of Geomemtric Sequences

Consider a geometric sequence \(a_1, a_2, \cdots, a_n\) with a common ration \(r\). The sum of its elements is obtained as follows

$$ S_n = a_1 + a_1r + a_1r^2 + \cdots + a_1r^{n - 1} $$

If we multiply by the common ratio \(r\):

$$ rS_n = a_1r + a_1r^2 + a_1r^3 + \cdots + a_1r^{n} $$

We substract both equations

$$ rS_n - S_n = a_1r^n - a_1 $$$$ S_n(r - 1) = a_1r^n - a_1 $$$$ S_n = \frac{a_1r^n - a_1}{r - 1}, r \neq 1 $$

The Sum of an Infinite Geomemtric Sequence

Given the formula for the sum of a geometric sequence

$$ S_n = \frac{a_1r^n - a_1}{r - 1} $$$$ S_n = \frac{-(a_1r^n - a_1)}{-(r - 1)} $$$$ S_n = \frac{a_1 - a_1r^n}{1 - r} $$$$ S_n = \frac{a_1}{1 - r} - \frac{a_1r^n}{1 - r} $$

In general, for values of \(r\) such that \(|r| < 0\), the expression \(r^n\) approaches zero as \(n\) gets larger and larger. Therefore

$$ S_n = \frac{a_1}{1 - r} - 0 $$

We say that the sum of the infinite geometric sequence is given by

$$ S_n = \frac{a_1}{1 - r}, |r| < 0 $$

If \(|r| > 1\), the absolute value of \(r^n\) increases without bound as \(n\) increases, thus \(|S_n|\) also increases without bound. Therefore we say that the sum of any infinite geometric sequence where \(|r| \geq 1\) does not exist.

Mathematical Induction