 # 2, 4, 6, 8, … a1, a2, a3, a4, … Arithmetic Sequences

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2, 4, 6, 8, … a1, a2, a3, a4, … Arithmetic Sequences
Sequence is a list of numbers typically with a pattern. 2, 4, 6, 8, … The first term in a sequence is denoted as a1, the second term is a2, and so on up to the nth term an. Each number in the list called a term. a1, a2, a3, a4, …

Finite Sequence has a fixed number of terms. {2, 4, 6, 8}
A sequence that has infinitely many terms is called an infinite sequence. {2, 4, 6, 8,…} Algebraically, a sequence can be written as an explicit formula or as a recursive formula. Explicit formulas show how to find a specific term number (n). Recursive formula show how to get from a given term (an-1) to the next term (an)

The common difference for the above arithmetic sequence is -3 .
An Arithmetic Sequence is a sequence where you use repeated addition (with same number) to get from one term to the next. Ex: 4, 1, -2, -5, … is an arithmetic sequence The number that needs to be added each time to get to the next term is called the common difference The common difference for the above arithmetic sequence is -3 .

Recursive formula for an Arithmetic Sequence:
Explicit formula for Arithmetic Sequence: an = dn + (a1 – d) Recursive formula for an Arithmetic Sequence: a1 = # an = an-1 + d Common difference First term For the example: 4, 1, -2, -5, … Explicit Formula Substitute the values: an = -3n + (4 - -3) So the explicit formula is: an = -3n + 7 The Recursive Formula is: a1 = 4 an = an-1 – 3

How do you add these sequences of numbers?
A series is the sum of ALL the terms of a sequence. (can be finite or infinite) A partial sum is the sum of the first n terms of a series…denoted Sn First term Last term Number of terms 𝑆 𝑛 = 𝑛 𝑎 1 + 𝑎 𝑛 2

1) Find S4. Find S20. For the example: 4, 1, -2, -5, …
𝑺 𝟒 = 𝟒(𝟒+ −𝟓) 𝟐 =−𝟐 Find S20. (If you don’t know the last term, use the explicit formula to find it!) Explicit formula is an = -3n + 7, so the 20th term is -3 (20) + 7 = -53 𝑺 𝟐𝟎 = 𝟐𝟎 (𝟒+ −𝟓𝟑) 𝟐 = −𝟒𝟗𝟎

Example: For the arithmetic sequence 2, 6, 10, 14, 18, …
Write the explicit formula for the sequence. Write the recursive formula for the sequence. c) Find the 15th partial sum of the sequence (S15). 𝑆 𝑛 = 𝑛 𝑎 1 + 𝑎 𝑛 2 a1 = # an = an-1 + d an = dn + (a1 – d)

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