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Wikipedia:Reference desk/Archives/Mathematics/2020 October 2

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October 2

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how is the #5 and #6 formulae of Arithmetic progression been derived?

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I wanted to know how is the #5 and #6 formulae of Arithmetic progression been derived. I know the derivation of or the nth term of an A.P. and the derivation of or the sum of nth terms of an A.P. — Preceding unsigned comment added by Huzaifa abedeen (talkcontribs) 10:04, 2 October 2020 (UTC)[reply]

The arithmetic mean of a bag of values is equal to their sum (given by #4) divided by the number of elements . So divide the formula at #4 by .
For an arithmetic progression starting at with increment , we have:
The equality can easily be formally proved by mathematical induction. Since is an arbitrary index, it is also the case that . Subtract these two equations from each other, and solve for .  --Lambiam 17:10, 2 October 2020 (UTC)[reply]
Derivation of or the arithmetic mean
Derivation of or the common difference
Thank you Lambian sir. You are a great mathematician. Huzaifa abedeen (talk) 09:34, 7 October 2020 (UTC) --Huzaifa abedeen[reply]