An arithmetic progression or arithmetic sequence (AP) is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2.
If the initial term of an arithmetic progression is a a and the common difference of successive members is d d, then the n n-th term of the sequence ( a n a_{n}) is given by:
a n = a + ( n ? 1 ) d {displaystyle a_{n}=a+(n-1)d},
If there are m terms in the AP, then a m a_{m} represents the last term which is given by:
a m = a + ( m ? 1 ) d {displaystyle a_{m}=a+(m-1)d}.
DU doesn't do subscripts any more, so see the link:
https://en.wikipedia.org/wiki/Arithmetic_progression#Derivation
The derivation is so clear and simple -- particularly when a specific example is shown -- that it impresses many non-math-inclined students to take a greater interest in math. I remember seeing the demonstration of the
sum of a geometric progression back in 7th grade -- it involves something called a
telescoping sum -- and it stuck with me so much that instead of memorizing the formula, I just repeat the derivation in my head when I need it. Strangely, I don't remember encountering the similar sum for an arithmetic series until 8th grade, when our geometry teacher (hat tip to Mr. Lengfield) asked us to sum the numbers 1 to 100 and then told us there is a quick trick to doing so. So I reinvented the quick trick (in slightly different form) for myself and came up with the answer 5,050 in only a few seconds. There is an anecdote that the great mathematician Gauss invented the same trick in elementary school, without anyone even suggesting there was a trick. He just knew there had to be, even then.
For the sake of completeness, here's a concrete example:
S = 1 + 2 + 3 + ... +100
S = 100 + 99 + 98 + ... +1
Add the two series together:
2S = (1+ 100) + (2+99) + (3+97) + ... + (100+1) {100 terms}
2S = 101 added together 100 times, or 101*100
and thus
S = 101*(100/2) = 5,050
This is conveniently described as the number of terms times the average of the terms (just by definition), the latter of which is the same as the average of the first and last terms , or
S = 100*(101/2)
which is obviously the same thing; easier to remember, less convenient for arithmetic.
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