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# Longest Arithmetic Progression

TopSequence a1, a2, a3, a4 ……. an is called an Arithmetic Progression if there exists a common difference 'd' between consecutive Numbers and this common difference is constant, a2 = a1 + d ,…. an = an-1 + d . Here a1 is first term, a2 is second term, a3 is third term, an is last term of series and d = second term – first term, (d = a2 - a1) of given series which is called common difference of Arithmetic progression. as longest arithmetic progression is sequence of n th term from end. Let there be an A.P with first term 'a' and common difference 'd', if there are 'm' terms in A.P.
then nth term from end = (m - n + 1) th term,
nth term from end = am-n + 1,
nth term from end = a +(m - n + 1 – 1)d,
nth term from end = a +(m – n) d,
Also if 'l' is the last term of A.P, then nth term from the end is the term of an A.P. whose first term is 'l' and common difference is '-d'.
N th term from end = last term + (n – 1)(-d),
N th term from end = l – (n - 1) d.
Suppose we have 6th term from the end of A.P. 17,14, 11 ,…..-40.
Then 'l' is last term = -40 and 'd' is common difference = -3, so 6th term from end = l – (6 - 1)d = -40 -5 x -3 = -25 this is the longest arithmetic progression.
We can solve other longest Arithmetic progression problems using above formula.