Sequence and Series: Formulae Explained

We may have learnt about sequence and series under different names at different grades.
Here are, it's formulae decoded.

Sequence

A sequence is an ordered list of numbers.
For Example, 1,2,3,4.........
                       2,5,8,11,......
                       100, 10, 1, 0.1, 0.01.......
It is classified in to Arithmetic Progression and Geometric Progression.

In Arithmetic Progression, each term of the progression is added or subtracted with a constant. i.e. Common difference "d".
The A.P thus formed is a, a+d, a+2d, a+3d........

To find the n^th term of A.P , we use the formula t_n = a-(n-1)d.
where, a is the first term, d is common difference, and n is the number of terms.
In case of first term, t₁ = a+0d
In case of second term, t₂ = a+1d
In case of third term, t₃ = a+2d
By these observations, we see that If it is a case of "nth term", Then the coefficient of d is n-1.
Hence, t_n = a+(n-1)d, is proved. We can take t_n as last term(l).
So that, the value of n can be derived:
l = a+(n-1)d
l -a = (n-1)d
(l -a)/d = n-1
((l-a)/d)-1 = n
i.e. n = l-a/d  +1

In Geometric Progression, each term of the progression is multiplied or divided with a constant i.e.Commom ratio "r".

The G.P. thus formed is a, ar, ar², ar³, .......

To find the nth term of G.P, we use the formula t_n = ar^n-1
where a is the first term, r is the common ratio, n is the number of terms
In case of first term, t₁= ar⁰
In case of second term, t₂ = ar¹
In case of third term, t₃ = ar²
By these observations, we see that If it is a case of "nth term", Then the power of r is n-1.
Hence, t_n = ar^n-1, is proved.

Series

A series is not but, the sum of the terms in the ordered list of numbers.
It is also classified into Arithmetic and Geometric Series.

In Arithmetic Series,each term of an A.P is added.
A.S thus formed is a + (a+d) + (a+2d) + (a+3d)+...........+ (a+(n-1)d)
To find the  sum of series with n terms, we use the formula S_n = n/2 (2a + (n-1)d)  or  n/2 (a+l)
where a is the first term, n is the number of term, d is the common difference, l is the last term of the A.S.

As said before, the last term of an A.S or A.P can be considered as the nth term. i.e tn = l
so that, (n/2) (a+1) = (n/2)(a+a+(n-1)d) = (n/2)(2a+(n-1)d). Hence, the two formulae are equal.
To derive the formula,
Sum of A.P, S_n = a + (a+d) + (a+2d) + (a+3d)+...........+ (a+(n-1)d)
                         =  a+a+d+a+2d+a+3d........+a+(n-1)d
                         = (a+a+a+a+a+a......+a) + (d+2d+3d+4d+.......+(n-1)d)
Since there are n times a,
                         = na + d(1+2+3+.......n-1)

to find the sum of 1+2+3+4.....+n, we can use n(n+1) / 2.
To get its derivation:
S_n = 1+2+3+4+......+(n-1)+n        >>1
We can write the following also like,
S_n = n+(n-1)+,,,,,,,,+4+3+2+1      >>2

Adding 1 and 2,
2S_n = (n+1) + (2+n-1) + .........+(n-1+2) + (n+1)
2S_n = n+1 + n+1 + ......+ n+1 + n+1
Since there are n times (n+1),
2S_n = n(n+1)
S_n = n(n+1) / 2

Coming back,
Sum  of A.P, S_n = na + d(n(n+1)/2  -n)       [Since n(n-1)/2 sums for n terms, and here it should be
                                                                       summed for (n-1) terms ]
                          = na + nd(n+1/ 2 -1)
                          = n(a + d(n+1-2/2 ))
                          = n(a + d(n-1 / 2))
                     S_n = n/2(2a + (n-1)d)
Hence, proved.


In Geometric Series,each term of a G.P is added.
The G.P. thus formed is a + ar +  ar² + ar³, .......+ ar^n-1
To find the sum of G.P with n terms, we use the formula S_n = a(rⁿ - 1) / r - 1, if r ≠ 1 and r ≠ 0
To derive the formula,
Sn = a + ar +  ar² + ar³, .......+ ar^n-1    >> 1

Multiplying r on both sides,
rS_n = ar +  ar² + ar³, .......+ arⁿ           >> 2

Subtracting 1 from 2,
rS_n - S_n = (ar +  ar² + ar³, .......+ arⁿ )- (a + ar +  ar² + ar³, .......+ ar^n-1 )  
S_n (r-1) = (ar +  ar² + ar³, .......+ arⁿ - a - ar -  ar² - ar³, .......- ar^n-1 )
S_n (r-1) = ( arⁿ - a)
S_n (r-1) = a(rⁿ - 1)
S_n  = a(rⁿ - 1) / r-1
Hence, proved.
If r = 1, S_n  = na [Since, the G.S would be a+a+a+a+a+......+a]
If r = 0, S_n  = a   [Since, the G.S would be a+0+0+0+0+0....+0]

Therfore, all formulae are explained.