Gauss Sequence?

A series is the indicated sum of all the terms in a sequence. An arithmetic series is the indicated sum of all the terms in an arithmetic sequence. One such example is

1 + 2 + 3 + 4 + . . . + 100.

The objective of this section is to find a formula for calculating the sum of the first n terms of any arithmetic sequence. As will be shown, the method we are using today was developed by the mathematician Carl F. Gauss (1777 – 1855) when he was a young student.

According to an old story, one day Gauss and his classmates were asked to find the sum of these first hundred counting numbers. All the other students in the class began by adding two numbers at a time, starting from the first term. This is a natural reaction. Also, it is a valid approach, although it is not the most efficient method, as will be seen. But Gauss found a quicker way. First, he wrote the sum twice, one in an ordinary order and the other in a reverse order:

1 + 2 + 3 + 4 + . . . + 99 + 100

100 + 99 + . . . + 4 + 3 + 2 + 1

By adding vertically, each pair of numbers adds up to 101:

1 + 2 + 3 + . . . + 98 + 99 + 100

100 + 99 + 98 + . . . + 3 + 2 + 1

101 + 101 + 101 + . . . + 101 + 101 + 101

Since there are 100 of these sums of 101, the total is 100Image101 = 10,100. Because this sum 10,100 is twice the sum of the numbers 1 through 100, we have:

1 + 2 + 3 + . . . + 98 + 99 + 100 = 100Image101 / 2 = 5050.

Using Gauss´ approach, we can find the formula for the sum of the first n terms of an arithmetic sequence having first term a1 and a common difference c, as follows:

Sn = a1 + a2 + a3 + . . . + an

= a1 + (a1 + c) + (a1 + 2c) + . . . + [a1 + (n-1)c]

Also, Sn = an + . . . + a3 + a2 + a1

= [a1 + (n-1)c] + . . . + (a1 + 2c) + (a1 + c) + a1

Therefore, 2Sn = (a1 + a2 + a3 + . . . + an)

+ (an + . . . + a3 + a2 + a1)

= [2a1 + (n-1)c] + . . . + [2a1 + (n-1)c]

= n [2a1 + (n-1)c]

Sn = n [2a1 + (n-1)c] / 2

= n [a1 + a1 + (n-1)c] / 2

= n [a1 + an] / 2

Generally, we use the Greek letter Sigma (sigma) to designate a series. The reason for using this sigma notation (summation symbol) is to reduce the amount of writing necessary. For example, the finite series 2 + 4 + 6 + 8 + 10 can be written as

Image

in which it is understood that k is to be replaced by each of the numbers 1, 2, 3, 4, and 5 in the expression 2k, and then the sum of the resulting values is to be indicated.

When k = 1,

k = 2,

k = 3,

k = 4,

k = 6, 2k = 2

2k = 4

2k = 6

2k = 8

2k = 10

Gauss Series

Hmm... any chance you are talking about the GAUSS programming language?

Yeah, give the next guy the points. The only question in my mind is, how did Gauss get credit for this? The formula to sum sequences like this was known way before Gauss - you're telling me that, just because some precocious kid comes up with it in class, he gets his name attached to it?

there's a undemanding huge difference of 6 the following you ought to count number out by technique of 6 to envision the size of the sequence, or, I choose, that 2 = 6n - at the same time as n =a million, 8 = 6n-4, at the same time as n =2 so 80 = 6n-4 at the same time as n = 14 The sequence is 14 words lengthy. in case you pair the first and extremely last time period you get 80 2, the 2d and thirteenth time period actually have a sum of 80 2. (you are able to ensure that mathematically, or with undemanding experience, it really is a linear sequence with a undemanding huge difference of 6). So there are 14/2 pairs of 80 2, or more advantageous perfect yet, the sum is 7x82.

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