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Using Summation Notation

Using Summation Notation for Arithmetic and Geometric Sequences top left image.Using Summation Notation for Arithmetic and Geometric Sequences top right image.
HomeAlgebraSequence and Series

Summation Notation is used to find the sum of some or all terms of a mathematical sequence. In this notation the symbol Σ, a stylized Sigma Greek letter, represents a summation math operation.

 n

ai

i = 1

This tells us that we are to sum the first n terms of the sequence (a1 + a2 + a3 + … + an). We also know to begin at the first term because of the index i = 1; i is a variable subscript, the index of summation.

As an example we evaluate:


 6

(i − 1) / i 2 = 0/1 + 1/4 + 2/9 + 3/16 + 4/25 + 5/36 = 3451/3600

i = 1


The index of summation can begin at any integer value:


 4

k3 = 23 + 33 + 43 = 8 + 27 + 64 = 99

k = 2

How to Sum the First n Terms of an Arithmetic Sequence

Since the first n terms of an Arithmetic Sequence can be written as:


a1, a1 + d, a1 = 2d, a1 + 3d, … , a1 + (n − 1)d

(Where d is the Common Difference)


Or,

a1, a1 + d, a1 + 2d, … , an − 2d, an − d, an


Then the sum of n terms can be found as follows:


sn = a1 + (a1 + d) + (a1 + 2d) + … + (an − 2d) + (an − d) + an


And, writing the sum backward:


sn = an + (an − d) + (an − 2d) + … + (a1 + 2d + (a1 + d) + a1


Adding both sn term by term we get:


2sn = (a1 + an) + (a1 + an) + (a1 + an) + (a1 + an) + (a1 + an) + (a1 + an)


Giving us 2sn = (n) (a1 + an) because there are n terms.


If 2sn = (n) (a1 + an)


Then sn must equal,


sn = (n) (a1 + an) / 2 where a1 is the first term and an is the nth term.

As an example we use the sequence set {5, 9, 13, 17}:


s = 05 + 09 + 13 + 17 = 44

s = 17 + 13 + 09 + 05 = 44

      22 + 22 + 22 + 22


2s = 22 x 4 = 88

s = 88 / 2 = 44


s4 = (n) (a1 + an) / 2 = 4 (05 + 17) / 2 = 44


It is easy to see that if we want to sum a large sequence of data the Arithmetic Sum Formula is quicker, much more efficient.

How to Sum the First n Terms of a Geometric Sequence

The first n terms of a Geometric Sequence can be written as:


a1, a1 r, a1 r2, … , a1 r n − 1

(Where r is the Common Ratio as defined on the Sequence and Series page)


So the sum of the first n terms is:

sn = a1 + a1 r + a1 r2 + … + a1 r n − 1


Then,


r sn = a1 r2 + a1 r3 + … + a1 r n


So,


sn − r sn = a1 + (a1 r − a1 r) + (a1 r2 − a1 r2) + … + a1 r n − 1 − a1 r n − 1) − a1 r n


Thus, (by factoring, sn − r sn, and adding the terms to the right of the equals)


sn (1 − r) = a1 − a1 r n


sn = a1 (1 − r n) / (1 − r)


As an example we evaluate the following geometric series with a first term of 5 and common ratio of −3:


s6 = a1 (1 − r n) / (1 − r) = 5 (1 − (−36 ) ) / (1 − (−3)) = 3650 / 4 = 912 1/2

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