I wanted to know how is the #5 and #6 formulae of Arithmetic progression been derived. I know the derivation of ${\displaystyle a_{n}}$or the nth term of an A.P. and the derivation of ${\displaystyle S_{n}}$or the sum of nth terms of an A.P. — Preceding unsigned comment added by Huzaifa abedeen (talk • contribs) 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 ${\displaystyle n}$. So divide the formula at #4 by ${\displaystyle n}$.

For an arithmetic progression starting at ${\displaystyle a_{1}}$with increment ${\displaystyle d}$, we have:

${\displaystyle a_{1}=a_{1},~a_{2}=a_{1}+d,~a_{3}=a_{1}+2d,~a_{4}=a_{1}+3d,~...,~a_{n}=a_{1}+(n-1)d.}$

The equality ${\displaystyle a_{n}=a_{1}+(n-1)d}$can easily be formally proved by mathematical induction. Since ${\displaystyle n}$is an arbitrary index, it is also the case that ${\displaystyle a_{m}=a_{1}+(m-1)d}$. Subtract these two equations from each other, and solve for ${\displaystyle d}$.  --Lambiam 17:10, 2 October 2020 (UTC)[reply]

Derivation of ${\displaystyle {\overline {a}}}$or the arithmetic mean

${\displaystyle S_{n}={\frac {n}{2}}(a_{1}+a_{n})}$

${\displaystyle \Rightarrow {\frac {S_{n}}{n}}={\frac {{\frac {n}{2}}(a_{1}+a_{n})}{n}}}$

${\displaystyle \Rightarrow {\overline {a}}={\frac {a_{1}+a_{n}}{2}}}$

Derivation of ${\displaystyle d}$or the common difference

${\displaystyle a_{m}-a_{n}={a_{1}+(m-1)d}-{a_{1}+(n-1)d}}$

${\displaystyle \Rightarrow a_{m}-a_{n}=(m-1)d-(n-1)d}$

${\displaystyle \Rightarrow a_{m}-a_{n}=d\{{(m-1)-(n-1)}\}}$

${\displaystyle \Rightarrow a_{m}-a_{n}=d\{m-n\}}$

${\displaystyle \Rightarrow d={\frac {a_{m}-a_{n}}{m-n}}}$

Thank you Lambian sir. You are a great mathematician. Huzaifa abedeen (talk) 09:34, 7 October 2020 (UTC) --Huzaifa abedeen[reply]