1.Write 4+6+8+10 in sigma notation:

I came up with sigma in the middle
3 on top
j=0 on the bottom

2. Find the 10th term of the sequence
-1/5,-1/20,-1/80 this was a geometric sequence so I said a(subscript10) = -1/5*1/4^(10-1)
=-1/5*(1/4)^9 = -1/5*1/2621444
= -1/1310702

To write the sum 4 + 6 + 8 + 10 in sigma notation, you start by determining the pattern or equation that generates each term in the sequence. In this case, the terms are consecutive even numbers starting from 4 and increasing by 2 each time. So, the general formula for the nth term is given by an = 2n + 2, where n represents the position of the term.

Now, to express the sum of these terms using sigma notation, you can write:

∑ (2n + 2), from j = 0 to 3.

Here, the sigma symbol (∑) indicates a summation, the variable j represents the position of each term in the series, and the range j = 0 to 3 indicates that we are adding up the terms where j takes the values 0, 1, 2, and 3. The term (2n + 2) is substituted using the formula we derived earlier.

To find the 10th term of the geometric sequence -1/5, -1/20, -1/80, we can use the formula for the nth term of a geometric sequence. The general formula is given by an = a1 * r^(n-1), where a1 is the first term, r is the common ratio, and n is the position of the term.

In this case, the first term is -1/5. The common ratio can be found by dividing any term by the previous term. For example, -1/20 divided by -1/5 is equal to 1/4.

Now, to find the 10th term, you substitute the given values into the formula:

a10 = -1/5 * (1/4)^(10-1)
= -1/5 * (1/4)^9
= -1/5 * 1/262144
= -1/131070.

So, the 10th term of the given geometric sequence is -1/131070.