1. Some numbers arranged in a definite order, according to a definite rule, are said to form a sequence.
2. 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., constant.
3. The constant number is called the common difference of the A.P.
4. If a is the first term and d is the common difference of an AP, then the general form of the AP is a, a + d, a + 2d, …
5. Let a be the first term and d be the common difference of an AP, then, its nth term or general term is given by
6. 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
7. If are in AP, then
(i) are in AP.
(ii) are in AP.
(iii) are in AP.
(iv) are in AP
8. Remember the following while working with consecutive terms in an AP.
(i) 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
(ii) 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
(iii) Five consecutive terms in an AP.
a – 2d, a – d, a, a + d, a + 2d
First term = a – 2d, common difference = d
9. The sum up to n terms of an AP whose first term is a and common difference d is given by
10. If the first term and the last term of an AP are and , then
(first term + last term)
If , the first term and , the last term, then