Revision Notes on Arithmetic Progression

If ‘a’ is the first term and ‘d’ is the common difference of the arithmetic progression, then its n^th term is given by a_n = a+(n-1)d
The sum, S_n of the first ‘n’ terms of the A.P. is given by S_n = n/2 [2a + (n-1)d]
If S_n is the sum of n terms of an A.P. whose first term is ‘a’ and last term is ‘l’,S_n = (n/2)(a + l)
If common difference is d, number of terms n and the last term l, then S_n = (n/2)[2l-(n -1)d]
If a fixed number is added or subtracted from each term of an A.P., then the resulting sequence is also an A.P. and it has the same common difference as that of the original A.P.
If each term of A.P is multiplied by some constant or divided by a non-zero fixed constant, the resulting sequence is an A.P. again.
If a₁, a₂, a₃, …, a_n andb₁, b₂, b₃, …, b_n, are in A.P. then a₁+b₁, a₂+b₂, a₃+b₃, ……, a_n+b_nand a₁–b₁, a₂–b₂, a₃–b₃, ……, a_n–b_n will also be in A.P.
Suppose a₁, a₂, a₃, ……,a_n are in A.P. then a_n, a_n–1, ……, a₃, a₂, a₁ will also be in A.P.
If n^th term of a series is t_n = An + B, then the series is in A.P.
If a₁, a₂, a₃, ……, a_n are in A.P., then a₁ + a_n = a₂ + a_n–1 = a₃ + a_n–2 = …… and so on.
In order to assume three terms in A.P. whose sum is given, they should be assumed as a-d, a, a+d.
Four terms of the A.P. whose sum is given should be assumed as a-3d, a-d, a+d, a+3d
Five convenient numbers in A.P. a–2b, a–b, a, a+b, a+2 b.
In general, we take a – rd, a – (r – 1)d, …., a – d, a, a + rd in case we have to take (2r + 1) terms in an A.P.
Likewise, any 2r terms of an A.P. should be assumed as: a – (2r-1)d, a – (2r – 3)d, …., a – d, a, a + d, ………….. , a+(2r-3)d, a + (2r-1)d.
The arithmetic mean of two numbers ‘a’ and ‘b’ is (a+b)/2.
The terms A₁, A₂, ….. , A_n are said to be arithmetic means between a and b if a, A₁, A₂, ….. , A_n, bis an A.P.
Clearly, ‘a’ is the first term, ‘b’ is the (n+2)^th term and ‘d’ is the common difference. Then, we have b = a+(n+2-1)d = a+(n+1)d