IMPORTANT TERMS, DEFINITIONS AND RESULTS

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., t_n + 1 - t_n= 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 t_n = a + (n - 1) d

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)
\Rightarrow nth term from the end = l – (n – 1) d

7. If a, b, c, are in AP, then
(i) (a + k), (b + k), (c + k) are in AP.
(ii) (a - k), (b - k), (c - k) are in AP.
(iii) ak, bk, ck, are in AP.
(iv) \dfrac{a}{k},\dfrac{b}{k},\dfrac{c}{k} are in AP (k\ne0)

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 S_n up to n terms of an AP whose first term is a and common difference d is given by S_n=\dfrac{n}{2}\ [2a+(n-1)d]

10. If the first term and the last term of an AP are t_1 and t_n, then
S_n=\dfrac{n}{2}(t_1+t_n)=\dfrac{n}{2} (first term + last term)
If t_1=a, the first term and t_n=l, the last term, then S_n=\dfrac{n}{2}(a+l)

11. S_n-S_{n-1}=t_n

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