0Sequence -
A succession of terms given by some numbers a1,a2,a3,... formed according to some rule or law.
Ex - 1,2,3,....
1/1,1/2,1/3,1/4,....
x,x^2,x^3,...
Series -
Series is the sum of the terms in the sequence.
Series can be given by -
Ex - 1/1 + 1/2 + 1/3 + .....
x,x^2,x^3,...
Arithmetic Progression (AP)
AP is a sequence where the difference between two successive terms is a constant value (d).
So, lets say the sequence a1,a2,a3,a4,a5,a6... be in an AP
So, a2-a1 = a3-a2 = a4-a3 = a5-a4 = a6-a5 = d
That's why terms in AP can be written as -
a, a+d, a+2d, a+3d,.....
where d is a common difference, and the first term is a
Here a1 = a, a2 = a+d, a3 = a+2d So, an (nth term) = a+(n-1)d
Sum of first n terms are given by Sn = n*(2a+(n-1)d)/2 or n*(a+l)/2 where l = an (nth term)
Note -
1. Sn - S(n-1) = an (nth term)
2. Sum of an AP is of the form An^2 +Bn
Properties of AP -
1. If each of the terms of an AP is increased, decreased, multiplied, or divided the resulting sequence is also an AP
2. If given in the question to let some(let's say x) numbers be in AP then choose terms as -
If x = 3 -> a-d, a, a+d (Here common diff = d)
If x = 4 -> a-3d, a-d, a+d, a+3d (Here common diff = 2d)
If x = 5 -> a-4d, a-2d, a, a+2d, a+4d (Here common diff = 2d)
....
3. Common difference can be 0,1,negative or positive
4. kth term from th last = (n-k+1)th term from the beginning so, a (kth from last) = a + (n-k+1 -1)d
5. If a, b, c are in AP => 2b = a+c
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