Derive The Formula Of Finding nth term from the end of an AP
Derivation for finding out the nth 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
nth term from the end of Arithmetic Progression = (p - n + 1)th term from the beginning
⇒ nth term from the end of Arithmetic Progression = A(p - n + 1) i.e. (p - n + 1)th term
⇒ nth term from the end of Arithmetic Progression = a + (p – n + 1 – 1)d
nth term from the end of Arithmetic Progression = a + (p - n).d
Also if 'l ' is the term of an Arithmetic Progression then nth term from end is the nth term of an Arithmetic progression whose first term is l and common difference is –d.
Now, nth term from the end of Arithmetic Progression = Last Term + (n - 1)(-d)
⇒ nth term from the end of Arithmetic Progression = l – (n - 1).d
Hope you get it!!