write the series with summation notation:-1,-7,-13,-19

each term decreases by six.

Ck= -1-k(6) k=0 to n

To write the series -1, -7, -13, -19 using summation notation, we first need to identify the pattern or rule that generates each term in the sequence. In this case, we notice that each term is obtained by subtracting 6 from the previous term.

To represent this pattern mathematically, we can use the formula for the general term of an arithmetic sequence. The general term of an arithmetic sequence is given by:

a_n = a_1 + (n-1)d

Where a_n represents the nth term, a_1 is the first term, n is the position of the term, and d is the common difference between consecutive terms.

In the given sequence, the first term (a_1) is -1, and the common difference (d) is -6 since we are subtracting 6 from each term.

Using the formula, we can rewrite the terms in the sequence as:

a_1 = -1
a_2 = -1 + (-6)(2-1)
a_3 = -1 + (-6)(3-1)
a_4 = -1 + (-6)(4-1)

Now, let's write the summation notation for this sequence. We will have the sum of n terms, where n represents the position of the last term we want to include.

The summation notation for this sequence is:

∑(from i = 1 to n) (-1 - 6(i-1))

Here, we are summing the expression -1 - 6(i-1) for each term from 1 to n. This notation represents the given sequence for any value of n.