SEQUENCES A sequence is a function whose domain in the set of positive integers. So if you have a function but limited the domain to the set of positive integers and you substituted them in beginning with 1 then 2 then 3 etc. you would generate a sequence. This is a sequence generated by putting 1, 2, 3, 4, 5 … in the function above.

We can look at the sequence and see a pattern: +3 For a sequence we usually use a little different notation than for functions. We use the letter a for a term in the sequence and subscript it with the term number. The term number is also the number that you substitute in the formula for a sequence to get that term’s value. This would be how we’d express the sequence above. I f we want to know the 10th term we’d put 10 in the sequence formula (often called the term generator)

Write the first five terms of the sequence. 2,5,10,17,26, Can you see the pattern and guess the next term? 37,

Can you figure out the formula for this sequence? We can see that each term is twice as big as the next except that the terms alternate signs. Alternating signs in a sequence mean there must be a negative sign to a power because odd powers would be negative and even powers would be positive. Let’s make a guess on what it might look like and then modify our guess until we have it right. Let’s sub in 1 then 2 then 3 and see what we get. Looks good but we lost the first term. Let’s modify our guess.

Another way to define a sequence is using what we call a recursive formula. Basically it is the rule of what you do to a term to get the term after it. +3 We saw the pattern in the first sequence: So to get to the next term we add three. The recursive formula would look like this: to get the nth termtake the n-1 term (the term right before it) and add 3

Let’s write the recursive formula for the other sequence we had: To get to the next term you multiply by negative two. With this formula, to get to the next term you must substitute the term before it. While this formula basically shows you the pattern, you can’t easily get at the 10th term without knowing the 9th term. You’d need the 9th term here to find the 10th term

Let’s try generating a sequence from a recursive formula: tells us the first term take the term number subtract the term before Use this formula to find the second term. would be when n = 2 1 = 1 Find the next 3 terms. 1, 1, 2, 2, 3, …

SUMMATION NOTATION Often we want to sum the terms in a sequence so summation notation is a short-hand way express this. This number tells us what integer to start subbing in to create the terms in a sequence This number tells us when to stop (the last integer to sub in). This is the formula to sub into This sign means to sum up each of the terms in the sequence

If the top number is an n, write out the sequence summing the terms with a … to the nth term: start by subbing in 0 for the k in the formula

Finally, let’s see if we can go backwards. Given the sum of the terms of the sequence, can we write it using summation notation? this appears to be the formula with a 12 subbed in Does 2k - 1 generate these terms subbing in 1 then 2 etc.?