Suppose, If you have a big series of numbers, and someone asked you the sum of the series, then what will you do? Will you add every number of the series one by one? I think this is not an effective and efficient way.

So, for solving these types of problems, we use SUMMATION. 

Let’s take the formula of summation and try to understand it.

  • n is the upper limit of summation, which told us about the stopping point of summation.
  • i is the index of summation which gives us the value for starting point of summation.
  • X is the expression which to is summed as if I say that write the sum of first 50 natural numbers, so x will be a natural number in this case.

Examples

  1. Find the sum of the first 10 natural numbers.

Sol.           i=110i=1+2+…+10

                              =1+2+3+4+5+6+7+8+9+10

  1. Find the sum of the 2n-1 sequence with upper point 3.

Sol. i=132n-1=2*1-1 +2*2-1+(2*3-1)

        = 9

These are the simple examples if you have a typical problem then,

  1. Find the sum of sequence i2 + i + 1 from 1 to 5 upper point





You can solve it by two methods:-

Method-1 

Simply you put it in summation formula and add expression terms

i=15i2 + i + 1 

But this a very lengthy method

 Method-2 

 We also have some results generated by using Arithmetic and Geometric Progressions.

 Arithmetic Progression

It is a type of sequence, in which the difference between two consecutive terms is constant.

This is known as the common difference of AP.

Example, 

3,9,12,15,18,… is an AP.

Its first term is 2 which is denoted by a. 

The common difference is 3, denoted by d and it is the same for every two consecutive terms, that’s why it is an AP.

 Geometric Progression

 It is a type of sequence, in which consecutive terms have a constant ratio. This constant ratio is called Common Ratio.

Example,

5,25,125,… Is a GP

a = 2

common ratio denoted by r is 2. It is the same for every consecutive term, that’s why it is a GP.

Sum Of AP and GP

Are you wondering, why we are talking about AP & GP in Summation?

Well, I want to inform you that, AP and GP has their formula for doing the sum of sequences. So, these two may help us in finding a summation of typical series.

  • Sum of AP Series:-

S = n22a+n-1d or S = n2a+l (sum of all terms of AP)



a = First term of Sequence

n = no of terms in the sequence

d = common difference of AP (d = a2 – a1 )

l = last term of the sequence

  • Sum of GP Series: –

            Sn = arn-1r-1 ( r ≠ 1, r > 1)   or    Sn = a1-rn(1-r) (r ≠ 1 , r <1)

a = First term of GP

n = no of terms in GP

r = common ratio of GP   (r = a2 / a1 )

If GP series is inifinite,         s=a1-r

 

General Results of sum of series.

  1. i=1ni= nn-12
  2. i=1ni2= nn+12n+16
  3. i=1ni3 = nn+122

  

Examples To Calculate Summations

 

            1.Find p=13p2

Sol.    Method 1:-

p=1np2= nn+12n+16

= 33+16+16

= 14

Method 2:-

p=13p2= 12 + 22 +32

    = 14





Let’s do some tricky questions based on summations.

  1. i=12(j=13(xj+ai)) where a = [1 0 -1] and b = [ 0 2 1]

Sol.   i=12(j=13(xj+ai))

  = i=12(j=13xj+j=13ai )

= i=12(x1+x2+x3+3ai)

  = i=12(0+2+1+3ai)

=i=12(3+3ai)

=i=12(3)+3i=12( ai)

=3(2) + 3 (a1+ a2+ a3)

=6 + 3 (1 + 0 + (-1) )

= 6 + 0

= 6

  1. Evaluate the following : –

i=11259

Sol.  i=11259=9+9+9+…+9  

(This Number (9) will be add 125 times, so there is a simple method for doing this and that is multiplication, which we have also done in the  above problem .)

.  i=11259=9×125

= 1125

16

  1. Show the following result using summation: –

p=1np+1p+5= 16n(n+7)(2n+7)

Sol.               L.H.S =  p=1np+1p+5

= p=1np2+6p+5

= p=1np2+p=1n6p+p=1n5

= 16n(n+1)(2n+1) + 6 × 12 n (n+1) +  5n

= 16n [ n+12n+1+18n+1+30]

= 16n [2n2+3n+1+18n+18+30]

= 16n[2n2+21n+49]

=16n(n+7)(2n+7)

Hence Proved.

 

Conclusion

Summation is a very useful technique to sum the larger expressions or series efficiently. It will also reduce the mistakes which can happen in calculating large series. We have also discussed Arithmetic and geometric progressions, which can be helpful in dealing with some type of series. We have also discussed the sum of Arithmetic progression and geometric progression and explained the summation with the working examples. The definition of the index of summation and the history related to summation has already been discussed.  By knowing about summation, you will be able, sum up, typical sequences and series as well.