# Arithmetic Progression

## Important Terms, Definitions and Results

• Some numbers arranged in a definite order, according to a definite rule, are said to form a sequence.
• A sequence is called an arithmetic progression (AP), if the difference of any of its terms and the preceding term is always the same.
i.e., an – an – 1 = constant.
• The constant number is called the common difference of the A.P.
• If a is the first term and d the common difference of an AP, then the general form of the AP is a, a + d, a + 2d, ...
• Let a be the first term and d be the common difference of an AP, then, its nth term or general is given by an = a + (n – 1) d
• If l is the last term of the AP, then nth term from the end is the nth term of an AP, whose first term is l and common difference is – d.
nth term from the end = Last term + (n – 1) (– d)
nth term from the end = l – (n – 1) d
• If a, b, c, are in AP, then
• (a + k), (b + k), (c + k) are in AP.
• (a – k), (b – k), (c – k) are in AP.
• ak, bk, ck, are in AP.
• a/k, b/k, c/k are in AP(k ≠ 0).
• Remember the following while working with consecutive terms in an AP.
• Three consecutive terms in an AP.
a – d, a, a + d
First term = a – d, common difference = d
Their sum = a – d + a + a + d = 3a
• Four consecutive terms in an AP.
a – 3d, a – d, a + d, a + 3d
First term = a – 3d, common difference = 2d
Their sum = a – 3d + a – d + a + d + a+ 3d = 4a
• Five consecutive terms in an AP.
a – 2d, a – d, a, a + d, a + 2d
First term = a – 2d, common difference = d
Their sum = a – 2d + a – d + a + a + d + a + 2d = 5a
• The sum Sn up to n terms of an AP whose first term is a and common difference d is given by
$S _ { n } = \frac { n } { 2 } [ 2 a + ( n - 1 ) d ]$
• If the first term and the last term of an AP are a and l, then
$S _ { n } = \frac { n } { 2 } \left( a + l \right) = \frac { n } { 2 } ( \text { first term } + \text { last term } )$
• an = Sn – Sn–1