Arithmetic Sequence

An Arithmetic Sequence is one in which there is a constant adder between terms. Some textbooks reference this as a sequence in which there is a constant difference between terms. An example of an Arithmetic Sequence is as follows:

3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43

In this particular sequence: the (value of the) first term is 3, the constant adder is 4, the (value of the) last term is 43, and the number of terms (in the sequence) is 11. The linkage between the first term "F", the constant adder "a", the last term "L", and the number of terms "N" is given by the formula:

L  =  F  +  A (N - 1)

Knowing any three of these variables should allow finding the fourth.

If the terms in an Arithmetic Sequence are added together  (3 + 7 + 11 + 15 + 19 + 23 + 27 + 31 + 35 + 39 + 43) ... usually referenced as an Arithmetic Series ... the sum of the terms "S" is given by the formula:

S  =  [ N ( F  +  L ) ]  / 2

For example, in the sequence above, the sum would be [ 11 * ( 3  +  43 ) ]  /  2  =  253. If you add the actual numbers contained in the series (3 + 7 + 11 + 15 + 19 + 23 + 27 + 31 + 35 + 39 + 43) you get the same result: 253.

Geometric Sequence

A Geometric Sequence is one in which there is a constant multiplier between terms. Most textbooks reference this as a sequence in which there is a constant ratio between the terms. An example of a Geometric Sequence is as follows:

3, 6, 12, 24, 48, 96, 192, 384

In this particular sequence: the (value of the) first term is 3, the constant multiplier is 2, the (value of the) last term is 384, and the number of terms (in the sequence) is 8. The linkage between the first term "F", the constant multiplier "M", the (value of the) last term "L", and the number of terms "N" is given by the formula:

L  =  F  * M(N-1)

(The last term is equal to the first term multiplied times the constant multiplier raised to the "N-1" power.) Knowing any three of these variables should allow finding the fourth.

If the terms in a Geometric Sequence are added together  (3 + 6 + 12 + 24 + 48 + 96 + 192 + 384) ... usually referenced as a Geometric Series ... the sum of the terms "S" is given by the formula:

S  =  [ F ( 1  -  MN ) ]  / [ 1 - M ]

For example, in the sequence above, the sum would be [ 3 * ( 1  -  28 ) ]  /  [ 1  -  2 ]  =  765. If you add the actual numbers contained in the series (3 + 6 + 12 + 24 + 48 + 96 + 192 + 384) you get the same result: 253.

One of the classic problems involving geometric sequences: A young lad approaches a businessman and asks for a job. The businessman indicates that he's not sure he needs any help. The lad suggests: "I'll work the first day for just a penny. On the second day I'll work for only 2 pennies. On the third day I'll work for 4 pennies ... etc.  All I ask is that you guarantee me 4 weeks ... 20 days ... of work."

Should the businessman accept the boys offer?

The sequence for pay is as follows:

$.01 , $.02, $.04, $.08, $.16, $.32, $.64 ... twenty total terms ... multiplier is 2.

The last term "L" = .01 * 219 = $ 5,242.88

The sum "S" of all twenty terms is = [ .01 * ( 1 - 220 ) ]  /  [ 1 - 2 ]  =  $ 10,485.75
 


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