Sequences+and+Series

Geometric and Arithmetic Sequences. The ansures to all your Geometric and Arithmetic problems __**Identifying a Geometric Sequences** __ Sequence B: 0.01, 0.06 , 0.36 , 2.16 , 12.96 , ...  Sequence C: 16, -8 , 4 , -2 , 1 , ... **
 * Sequences of numbers that follow a pattern of multiplying a fixed number from one term to the next are called geometric sequences. The following sequences are geometric sequences: **
 * Sequence A: 1, 2 , 4 , 8 , 16 , ...
 * For sequence A, if we multiply by 2 to the first number we will get the second number. This works for any pair of consecutive numbers. The second number times 2 is the third number: 2 × 2 = 4, and so on. **
 * For sequence B, if we multiply by 6 to the first number we will get the second number. This also works for any pair of consecutive numbers. The third number times 6 is the fourth number: 0.36 × 6 = 2.16, which will work throughout the entire sequence. **
 * Sequence C is a little different because it seems that we are dividing; yet to stay consistent with the theme of geometric sequences, we must think in terms of multiplication. We need to multiply by -1/2 to the first number to get the second number. This too works for any pair of consecutive numbers. The fourth number times -1/2 is the fifth number: -2 × -1/2 = 1.

** ** Generic Sequence: a1, a2, a3, a4, ... ** To find out more information about Geometric Sequences go to [|MATHguide.com]   ** __Identifying a Arithmetic Sequences__
 * Because these sequences behave according to this simple rule of multiplying a constant number to one term to get to another, they are called geometric sequences. So that we can examine these sequences to greater depth, we must know that the fixed numbers that bind each sequence together are called the common ratios. Mathematicians use the letter //r// when referring to these types of sequences. **
 * Mathematicians also refer to generic sequences using the letter //a// along with subscripts that correspond to the term numbers as follows: **
 * This means that if we refer to the tenth term of a certain sequence, we will label it a10. a14 is the 14th term. This notation is necessary for calculating nth terms, or an, of sequences. **
 * // r //**** can be calculated by dividing any two consecutive terms in a geometric sequence. The formula for calculating //r// is ****where n is any positive integer greater than 1. 
 * Sequences of numbers that follow a pattern of adding a fixed number from one term to the next are called arithmetic sequences. The following sequences are arithmetic sequences:**

Sequence B: 26, 31 , 36 , 41 , 46 , ... Sequence C: 20, 18 , 16 , 14 , 12 , ...**
 * Sequence A: 5, 8 , 11 , 14 , 17 , ...

To find more information about Arithmetic Sequences, go to [|MATHguide.com]** 
 * For sequence A, if we add 3 to the first number we will get the second number. This works for any pair of consecutive numbers. The second number plus 3 is the third number: 8 + 3 = 11, and so on.**
 * For sequence B, if we add 5 to the first number we will get the second number. This also works for any pair of consecutive numbers. The third number plus 5 is the fourth number: 36 + 5 = 41, which will work throughout the entire sequence.**
 * Sequence C is a little different because we need to add -2 to the first number to get the second number. This too works for any pair of consecutive numbers. The fourth number plus -2 is the fifth number: 14 + (-2) = 12.**
 * Because these sequences behave according to this simple rule of addiing a constant number to one term to get to another, they are called arithmetic sequences. So that we can examine these sequences to greater depth, we must know that the fixed numbers that bind each sequence together are called the common differences. Sometimes mathematicians use the letter //d// when referring to these types of sequences.**
 * Mathematicians also refer to generic sequences using the letter //a// along with subscripts that correspond to the term numbers as follows:**
 * Generic Sequence: a1, a2, a3, a4, ...**
 * This means that if we refer to the fifth term of a certain sequence, we will label it a5. a17 is the 17th term. This notation is necessary for calculating nth terms, or an, of sequences.**
 * //d// can be calculated by subtracting any two consecutive terms in an arithmetic sequence.**
 * d = an - an - 1, where n is any positive integer greater than 1.

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