Factorials and Sequences Sequences and Series Factorials and Sequences

Factorials Factorials are numbers that are found by using the products of consecutive numbers. What does that mean? 1 * 2 * 3 * 4 = 24 In factorial notation, this would be 4! Try one: 7! 5! 3! Now use the calculator. Press the number. (7) <Math>; Arrow over to PRB; Option 4 <Enter> (5040)

Factorials Remember 7! = 1 * 2 * 3 * 4 * 5 * 6 * 7 We can use this information to simplify. 8! 5! = 336 1∗2 ∗ 3 ∗ 4 ∗ 5 ∗ 6 ∗7 ∗ 8 1 ∗ 2 ∗ 3 ∗ 4 ∗ 5 6 * 7 * 8 336 Try a couple. 6! 4! 9! 7! 12! 8! 5! 8! 30 72 11880 1 336 Expand the factorial. Cancel out the numbers that are the same. Multiply what is left.

Sequences Sequence means to follow one thing after another in a certain order. Same thing in math. We need to know the notation of sequence in order to work with it. 𝑎 𝑛 = 2n – 1 is a sequence. 𝑎 𝑛 is the name of the sequence 2n – 1 is the operation needed to satisfy the sequence. Notice the “n”? It is the same number for both.

Sequences Find the first 4 terms of the sequence. (That means n = 1; n = 2; n = 3; n = 4) 𝑎 𝑛 = 2n – 1 𝑎 1 = 2(1) – 1 = 2 – 1 = 1 𝑎 2 = 2(2) – 1 = 4 – 1 = 3 𝑎 3 = 2(3) – 1 = 6 – 1 = 5 𝑎 4 = 2(4) – 1 = 8 – 1 = 7 Plug each of the n values into the sequence and solve. (1, 3, 5, 7)

Sequences 𝑎 𝑛 = (−1) 𝑛 𝑛+1 Try a harder one: 𝑎 1 = (−1) 1 1+1 = −1 2 𝑎 𝑛 = (−1) 𝑛 𝑛+1 𝑎 1 = (−1) 1 1+1 = −1 2 𝑎 2 = (−1) 2 2+1 = 1 3 𝑎 3 = (−1) 3 3+1 = −1 4 𝑎 4 = (−1) 4 4+1 = 1 5 ( −1 2 , 1 3 , −1 4 , 1 5 )

Sequences Sometimes you only need to find one term. You know which one based on the “n” value asked for. 𝑎 𝑛 = 𝑛 (𝑛+1) 2 , 𝑎 15 = ? 𝑎 15 = 15 (15+1) 2 = 15 16 2 = 15 256