Derive The Formula Of Finding nth term from the end of an AP

**Derivation for finding out the ****n**^{th}** term of a.p from its End:**

Consider an Arithmetic progression having first term *a* and common difference *d*.

Let us suppose that there are *p* terms in the A. P., then*n*^{th} term from the end of Arithmetic Progression = (*p* - *n* + 1)^{th} term from the beginning

⇒ *n*^{th} term from the end of Arithmetic Progression = *A*_{(p - n + 1)} i.e. (*p* - *n* + 1)^{th} term

⇒ *n*^{th} term from the end of Arithmetic Progression = *a* + (*p – n* + 1 – 1)*d*

*n*^{th} term from the end of Arithmetic Progression = *a* + (*p* - *n*).*d*

Also if '*l *' is the term of an Arithmetic Progression then *n*^{th} term from end is the *n*^{th} term of an Arithmetic progression whose first term is *l* and common difference is –*d*.

Now, *n*^{th} term from the end of Arithmetic Progression = Last Term + (*n *- 1)(-*d*)

⇒ *n*^{th} term from the end of Arithmetic Progression = *l* – (*n *- 1).*d*

Hope you get it!!

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