How is the sum expressed in sigma notation?

11 + 17 + 23 + 29 + 35 + 41

Solution:

Given that,

11 + 17 + 23 + 29 + 35 + 41

We have to express the sum in sigma notation

Analyse the series

5 + 6(1) = 11

5 + 6(2) = 17

5 + 6(3) = 5 + 18 = 23

5 + 6(4) = 5 + 24 = 29

5 + 6(5) = 5 + 30 = 35

5 + 6(6) = 5 + 36 = 41

Thus the series goes on like this:

5 + 6(n) , where n = 1 to 6

This can be expressed in sigma notation as:

We can expand the sigma notation and verify the results

The given sum is expressed by the notation, ∑ 11 + 6n, where the lower limit is n = 0 and the upper limit is n =5.

Step-by-step explanation:

Step 1; First, we need to determine how the numbers in the series (11 + 17 + 23 + 29 + 35 + 41) are related to one another. The initial value is 11 and for every successive number, there is a difference of 6.

First number = 11,

Second number = 11 + 6 = 11 + (6) × 1 = 17,

Third number = 11 + 6 + 6 = 11 + (6) × 2 = 23,

Fourth number = 11 + 6 + 6 + 6 = 11 + (6) × 3 = 29,

Fifth number = 11 + 6 + 6 + 6 + 6 = 11 + (6) × 4 = 35,

Sixth number = 11 + 6 + 6 + 6 + 6 + 6 = 11 + (6) × 5 = 41.

Step 2; So for the sigma notation, we insert the initial term i.e. 11 and to represent the term that is added, we put 6n.

So by substituting the values in the notation, ∑ 11 + 6n, where the lower limit is n = 0 and upper limit is n =5, we get;

n = 0,  11 + 6n = 11 + (6) × 0 = 11,

n = 1,  11 + 6n = 11 + (6) × 1 = 17,

n = 2,  11 + 6n = 11 + (6) × 2 = 23,

n = 3,  11 + 6n = 11 + (6) × 3 = 29,

n = 4,  11 + 6n = 11 + (6) × 4 = 35,

n = 5,  11 + 6n = 11 + (6) × 5 = 41.

The value of ∑ for all these values is given by 11 + 17 + 23 + 29 + 35 + 41 = 156.