XXVI Sequences

Series of successions of numbers that are formed by a common law, meaning each number is formed by the same arithmetic operation on the former.

Important aspects of sequences or progressions are:

• The law of its formation
• The sum of a given number of its terms

Arithmetic Progressions

Are sequences formed by addition or subtraction of each number with a fixed constant, called the common difference of the series.

Examples

1) $7,13,19,25,\text{\textellipsis}$, common difference: $6$
2) $6,4,2,0,-2,04,\text{\textellipsis}$, common difference: $-2$

Therefore if $a, b, c$ are in arithmetic sequence:

$b - a = c - b$

That means, subtracting a number by its predecessor yields the common difference.

Describing any Term in the Sequence

if $a = \text{first term of a series}$ and $d = \text{common difference}$:

$a, a+d, a+2d, a+3d,\text{\textellipsis}$

This implies the formula for the $n$th term of the progression:

$a + (n-1)d$

Last Term of a Sequence

$l = a+d(n-1)$

Sum of any Number of Terms

$n = \text{number of terms whose sum is required}$
$s = \text{sum of n terms}$

$s = \frac n 2 (2a+d(n-1))$

Arithmetic Mean

Given three numbers in any arithmetic progression, the middle term is always their arithmetic mean.

$b - a = c - b$
$2b = a+c$
$b = \frac{a+c}{2}$

Harmonic Progressions

Are series of reciprocals:

If $1,3,5,7,\text{\textellipsis}$ is a series in arithmetic progression, then

$1, \frac 1 3, \frac 1 5, \frac 1 7, \text{\textellipsis}$ is a series in harmonic progression.

Sums of harmonic sequences can be found using their arithmetic counter parts.

Harmonic Mean

The mean of a harmonic series is found using the corresponding arithmetic sequence, and finding the reciprocal of the arithmetic mean.

Geometric Progressions

Are series where each number is a multiple of its preceding number and a common ratio.

Rule

If $a,b,c,\text{\textellipsis}$ is a series in geometric progression:

$\frac b a = \frac c b$

So each term over the previous one is the same ratio.

$a = \text{first term}$
$r = \text{common ratio}$

$a,ar,ar^2,ar^3,\text{\textellipsis}$

Note that the logarithms of a geometric sequence form a series in arithmetic progression:

$\log a, \log a + \log r, \log a + 2\log r, \log a + 3\log r,\text{\textellipsis}$

Formula for the $n$th term of a geometric series:

$x = ar^{n-1}$

Geometric Mean

$\frac b a = \frac c b$

$b^2 = ac$
<?ktc b = \sqrt{ac} ?>

Sum of any Number of Terms

If $r > 1$:

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

Else ($r < 1$):

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

Increasing Sequences

With a common ratio of $1$ or greater the sum of a geometric progression increases with the number of terms $n$.

Therefore if $n$ increases without limit, so does their sum, meaning both increase towards infinity.

Decreasing Sequences

If $r < 1$ as $n$ grows towards infinity the value of the terms becomes increasingly low - they decrease indefinitely. Therefore there is a limit to their sum.

Consider that $\frac 1 9 = 0.11\text{=1}$ is in effect a geometric sequence:

$\frac {1}{10} + \frac {1}{10^2} + \frac {1}{10^3} + \text{\textellipsis}$

So no matter how large $n$ may become, the sum $S_n$ never exceeds $\frac 1 9$.

The Sum to a Limit

$\frac 1 2 + \frac {1}{2^2} + \frac {1}{2^3} + \text{\textellipsis}$

The sum approaches the limit $1$ as $n$ increases towards infinity.

Therefore given a ratio $r < 1$, the following formula calculates the limiting sum of a sequence:

$S_{\infin} = \frac {a}{1-r}$