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# Sequence and series

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### Sequence and series

1. 1. Arithmetic and Geometric
2. 2. Theorem: The sum of the integers from 1 to n is given by the formula: ( )1 2 1 += nnSn
3. 3. Arithmetic Sequence It is a sequence in which the difference between consecutive terms is constant and has the form: Note: An arithmetic sequence exhibit constant growth ( )dnadadaa 1,,2,, 1111 −+++ 
4. 4. Arithmetic Sequence Examples: 1, 4, 7, 10, 13, … 6, 11, 16, 21, 26, … 14, 25, 36, 47, 58, … 4, 2, 0, -2, -4, … -1, -7, -13, -19, -25, …
5. 5. Arithmetic Sequence General Term: ( )dnaan 11 −+=
6. 6. Arithmetic Series A series is an indicated sum of terms of a sequence. If the terms form an arithmetic sequence with first term a1 and common difference d, the indicated sum of terms is called an arithmetic series. The sum of the first n terms, represented as Sn, is
7. 7. Formula nnn aaaaaS +++++= −1321 
8. 8. Arithmetic Series Let Sn = a1 + a2 + … + an be an arithmetic series then ( ) 2 1 n n aan S + =
9. 9. Arithmetic Series Let Sn = a1 + a2 + … + an be an arithmetic series with constant difference d, then: ( )[ ] 2 12 1 dnan Sn −+ =
10. 10. Geometric Sequence (Or Geometric Progression) is a sequence in which each term after the first is obtained by multiplying the preceding term by a common multiplier. The common multiplier is called the common ratio of the sequence.
11. 11. Geometric Sequence Examples: 1, 2, 4, 8, 16, … 8, 4, 2, 1, ½, … 3, 9, 27, 81, 243, … 1, -4, 16, -64, 256, …
12. 12. Geometric Series A geometric series is the indicated sum of a geometric sequence. The following are examples of geometric series: 1 + 2 + 4 + 8 + … 8 + 4 + 2 + 1 + … 1 + (-4) + 16 + (-64) + …
13. 13. Geometric Series The sum of the first n terms of a geometric sequence that has a first term a1 and a common ration r is given by: ( ) .1anyfor, 1 11 ≠ − − = r r ra S n n