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.


1) 7,13,19,25,7,13,19,25,\text{\textellipsis}, common difference: 66
2) 6,4,2,0,2,04,6,4,2,0,-2,04,\text{\textellipsis}, common difference: 2-2

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

ba=cbb - a = c - b

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

Describing any Term in the Sequence

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

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

This implies the formula for the nnth term of the progression:

a+(n1)da + (n-1)d

Last Term of a Sequence

l=a+d(n1)l = a+d(n-1)

Sum of any Number of Terms

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

s=n2(2a+d(n1))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.

ba=cbb - a = c - b
2b=a+c2b = a+c
b=a+c2b = \frac{a+c}{2}

Harmonic Progressions

Are series of reciprocals:

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

1,13,15,17,1, \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.


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

ba=cb\frac b a = \frac c b

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

a=first terma = \text{first term}
r=common ratior = \text{common ratio}


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

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

Formula for the nnth term of a geometric series:

x=arn1x = ar^{n-1}

Geometric Mean

ba=cb\frac b a = \frac c b

b2=acb^2 = ac
<?ktc b = \sqrt{ac} ?>

Sum of any Number of Terms

If r>1r > 1:

Sn=a(rn1)r1S_n = \frac{a(r^n-1)}{r-1}

Else (r<1r < 1):

Sn=a(1rn)1rS_n = \frac{a(1-r^n)}{1-r}

Increasing Sequences

With a common ratio of 11 or greater the sum of a geometric progression increases with the number of terms nn.

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

Decreasing Sequences

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

Consider that 19=0.111ˉ\frac 1 9 = 0.11\text{=1} is in effect a geometric sequence:

110+1102+1103+\frac {1}{10} + \frac {1}{10^2} + \frac {1}{10^3} + \text{\textellipsis}

So no matter how large nn may become, the sum SnS_n never exceeds 19\frac 1 9.

The Sum to a Limit

12+122+123+\frac 1 2 + \frac {1}{2^2} + \frac {1}{2^3} + \text{\textellipsis}

The sum approaches the limit 11 as nn increases towards infinity.

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

S=a1rS_{\infin} = \frac {a}{1-r}