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Arithematic Progressions Upto nth term
Arithematic Progressions Upto nth term
1. Sequence :
In mathematics a sequence is an ordered list of objects (or events or numbers)
Note: Exactly the same elements can appear multiple times at different positions in the sequence.
Examples:
i) 1,3,5,7,9,.........
ii) 1,1,2,2,3,3,3,.........
2. Finite and Infinite sequence:
i) A sequence is said to be finite if it has finite number of terms.
Examples:
Sequence of first five prime numbers
2,3,5,7,11
ii) A sequence is said to be infinite if it has infinite number of terms.
Example :
Sequence of all even natural numbers
2,4,6,8,...........
3. Series:
By adding or subtracting the terms of a sequence, we get an expression which is called a series. i.e If is a sequence, then is a series.
Example:
i) 1+2+3+4+...............+n
ii) 2-4+6-8+10-.......... -
Arithematic Progressions Upto nth term
Arithematic Progressions Upto nth term
Progression:
It is not necessary that the terms of a sequence always follow a cercation pattern or they are described by some explicit formula for the nth term. Those sequences whose terms follow certain patterns are called progression
Example :
i) 2,5,8,11,.............
ii)\(\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{{16}},...........\)
(iii) \(\frac{1}{3},\frac{1}{5},\frac{1}{7},\frac{1}{9}\),.........
Arithmetic Progression (A.P)
Arithmetic progression is a sequence whose terms increase or decrease by a fixed number. this fixed number is called common difference denoted by 'd'.
Examples :
i) 2,5,8,11,14,........
ii) -2,-3,-4,-6,.............…
General term (or) nth term (or) Last term of an A.P
Consider an A.P 2,5,8,11,............. here except the first term 2, every term is obtained by adding a fixed number 3 to its previous number.
Now, first term = a=2
common difference d= 5- 2 =3
Now, First term = t1 =2 = a
Second term = t2 = 5= 2+1.3
Third term = t3 = 8 = 2+2.3
Fourth term t4 = 11 = 2 +3.3
Proceeding in this way we will get nth term =tn = a+(n-1)d.
7. In an A.P we take the terms as follows
t1 =a, t2 = a+d, t3 = a+2d, tn = a+(n-1)d.
8. If '\(l\) ' is the last term and 'd' be common difference of an A.P, then mth term from the end is
Note :
mth term from the end = (n -m+1)th term from the beginning
9. Selection of terms in an A.P
10. Some facts about an A.P
i) a,b,c are in \(A.P \Leftrightarrow 2b = a + c\)
ii) In a finite A.P the sum of the terms equidistant from the begining and the end is always same and is equal to the sum of the first and last term
i.e \({a_1} + {a_n} = {a_2} + {a_{n - 1}} = {a_3} + {a_{n - 2}}\)
iii) If a1,a2,a3,................an are in A.P, then an, an-1, an-2, .............a3, a2,a1 are in A.P
iv) \({a_1} \pm \lambda ,{a_2} \pm \lambda ,{a_3} \pm \lambda ,..........{a_n} \pm \lambda \) are in A.P, where \(\lambda \in R\)
v)\(\lambda {a_1},\lambda {a_2},\lambda {a_3},..........\lambda {a_n}\) are in A.P where \(\lambda \in R - \{ 0\} \)
vi) If pth term of an A.P is 'q' and qth term is 'p' then Tp+q = 0
vii)If \({a_1},{a_2},{a_3}............{a_n}\) and \({b_1},{b_2},{b_3}............{b_n}\) are two A.P's then \({a_1} \pm {b_1},{a_2} \pm {b_2},{a_3} \pm {b_3}............\) are in A.P, but \({a_1}{b_1},{a_2}{b_2}\) ........and \(\frac{{{a_1}}}{{{b_1}}},\frac{{{a_2}}}{{{b_2}}}.........\) are not in A.P
viii)If the terms of an A.P are chosen at regular intervals, then they form an A.P
ix) If a constant 'k' is added to each term of A.P with common differena 'd', then the resulting sequence also will be in A.P, with common difference (d+k)
x) If every term is multiplied by a constant 'k' then the resulting sequence will also be in A.P, wit the first term 'ka' and common difference 'kd'.
xi) If nth term of the sequence Tn=An+B i.e linear expression in n, then the sequence is A.P with first term is A+B and common difference A (i.e., coefficient of n).