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# Arithmetic series formula

The sum of the first n terms in an arithmetic sequence is (n/2)⋅(a₁+aₙ). It is called the arithmetic series formula. Learn more about it here. Created by Sal Khan.

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• At mins and after, I understand what you did, I don't understand why. Do we need to write every series backwards every time we want the equation for the sequence or is there a better way to go about this? And where did writing it backwards come from?
• No need to write all that out every time. The purpose of all that is to illustrate why the formula works. The fundamental insight that originally led to the creation of this formula probably started with the observation that the sum of the first term and last term in an arithmetic series is always the same as the sum of the 2nd and 2nd-to-last, 3rd and 3rd-to-last, etc. Try it in your head with a simple series, such as whole numbers from 1 to 10, to see what I mean. Writing a sequence forward then backward, one above the other, is just another way to look at that same phenomenon because it aligns the first term with the last term, 2nd term with 2nd-to-last, etc. It also leaves no confusion about what to do with the middle term when there are an odd number of terms.
• Why does Sal swap the order of the sequence and add it to itself ? I understand that it works in finding the series, but I don't understand why it works.
• Imagine the sequence of whole numbers from 1 to 10 written out. Then imagine the same sequence written in reverse order just below the first. When you add the vertical pairs of corresponding terms, you will get the same result each time, which in this example is 11 (1+10=11, 2+9=11, 3+8=11 ...). This is because as you move from one pair to the next, the upper term increases by the same amount that the lower term decreases. Sal is illustrating this principle in a general form, showing how it applies to ANY arithmetic sequence.
• What is the general formula for geometric series? If there is one.
• Geometric series:
finite geometric series- Sn= a sub 1(1-r^n) / 1-r
infinite geometric series- Sn= a sub 1 / 1-r
• Sorry about this question but, what is the difference between a arithmetic sequence and a arithmetic series. I can't seem to wrap my brain around this topic.
• series 1+2+3+4...
sequence 1,2,3,4...
literally the only difference is being separated by commas and plus signs
• Just for example: What if we have sigma notation?Σ It has lower limit(a) as 1, upper limit as 4, then,Σ, a(a-1).
• What exactly is "a"? Like, when in the equation a + d(n-1) ? What number in the sequence is "a"??
• A is the first term of a sequence, you may also see it written as t1 (t sub 1).
• In an infinite arithmetic series, how can you do the average of the terms ?
• there is no average, it just keeps getting bigger and bigger
• What is the easiest way to see if a sequence is arithmetic or geometric and, similarly, to see if a series is one or the other?
• Consider the sequence of numbers we will represent as A, B, C, D, E .....
If B-A = C-B = D-C = E-D.... then it is arithmetic.
If A/B = B/C = C/D = E/D .... then it is geometric.
An arithmetic series is the sum of an arithmetic sequence
A geometric series is the sum of a geometric sequence.
Thus, with the series you just see if the relationship between the terms is arithmetic (each term increases or decreases by adding a constant to the previous term ) or geometric (each term is found by multiplying the previous term by a constant).
• at how did
1d + (n-2)d become (n - 2 + 1)d?
• It's an arithmetic factoring, you can see it more clearly if you you first expand the terms in the parenthesis and then take our the common `d` term:
``1d + (n - 2)d1d + nd - 2d(1 + n - 2) d(n - 2 + 1) d``