Series Adding terms of a sequence (11.4)

Add sequence Our first arithmetic sequence: 2, 7, 12, 17, … What is the sum of the first term? The first two terms? The first three terms? First four? Five?

Add sequence Our first arithmetic sequence: 2, 7, 12, 17, … What is the sum of the first term? 2 The first two terms? 9 The first three terms? 21 First four? 38 Five? 60 Do you see what we’re doing?

A series If you add the terms of a sequence, you get a series. Since sequences are an infinite string of elements (we’re using numbers), a series of a sequence would be infinitely big. That’s a mess. So, mostly we think about partial series or partial sums, the sum of just part of the terms in a sequence. They are also known as finite series.

Summation notation Adding a bunch of terms can take a bunch of paper, so we use a simpler symbol to show we’re adding those terms. Sometimes we use S to indicate the “sum.” S 9 would mean the sum of the first 9 terms. Find S 9 for our POD sequence.

Summation notation Adding a bunch of terms can take a bunch of paper, so we use a simpler symbol to show we’re adding those terms. Sometimes we use something called summation notation, and we use the capital Greek letter sigma: Σ We can include more information with this than with S notation.

Summation notation Let’s see the parts. In this notation, we start adding from the 1 st to the 10 th terms. We see this by looking above and below the sigma. The statement to the side actually tells us what the terms are. Plug in 1 to get the first term. Plug in 2 to get the second term, etc.

Summation notation Let’s see the parts. In this notation, we start adding from the 1 st to the 10 th terms. We see this by looking above and below the sigma. The statement to the side actually tells us what the terms are. Plug in 1 to get the first term. Plug in 2 to get the second term, etc.

A little more complicated What are the terms we’re adding? What’s the sum? How could we notate adding the third through the sixth terms?

A little more complicated What are the terms we’re adding? What’s the sum? 91 Third through sixth terms?

A sum of mixed operations Just like before, plug in 1 for the first term, etc. What kind of sequence is this?

A sum of mixed operations Just like before, plug in 1 for the first term, etc. It’s an arithmetic sequence.

Ooh, exponents Same process… What kind of sequence is this?

Ooh, exponents Same process… It’s a geometric sequence.

Make up one of your own Using summation notation, determine how many terms you want to add, and the formula for those terms. Write it using correct summation notation. Figure out the sum. Have a partner check your work. Turn it into the homework folder.