Lesson Arithmetic Progression

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1. Progression means process of progressing (a movement forward) i.e., a series of numbers or quantities in which there is always the same relation between each quantity and the one succeeding it. 
2.  Arithmetic Progression (A.P): A sequence in which each term is obtained by adding some constant to the preceding term except for the first term is said to be in Arithmetic Progression.  That is the difference between two successive terms is same called �common difference� (c.d).
e.g. (i) 1, 5, 9, 13, ��      (ii)  - 10, - 7,  - 4, - 1, 2, ���      (iii) 5, 5, 5, 5, 5, ���..

3.  If t1, t2, t3, t4, ��� are the terms of an A.P., then t2 � t1 = t3 � t2 = t4 � t3 = ����= tn � tn-1

4.  If �a� is the first term and �d� is the common difference then the terms of an A.P. are a, a + d, a + 2d, a + 3d, a + 4d, �����.

5.  The general term (or) nth term of an A.P. is, tn = a + (n � 1)d
e.g. 1, 4, 7, 10, ���.., find t10 and tn
 a = 1, d = t2 � t1 = 4 � 1 = 3,  t10 = 1 + (10 � 1)(3) = 1 + 27 = 28; tn = 1 + (n - 1)(3) = 1 + 3n � 3 = 3n � 2
e.g. x, 3x � 1, 5x � 2, 7x � 3, ���.., find t12 
d = t2 � t1 = (3x � 1) � x = 2x � 1;  t12 = x + (12 � 1)(2x � 1) = x + 22x � 11 = 23x � 11

6.  Check the sequences (i) 2, 6, 12, 20, 30, ��..  (ii) 1, 3, 9, 27, ��..  (iii) x, 2x^2, 3x^3, 4x^4, �..   are in Arithmetic Progression ? Can you find the next terms in those sequences?
key: The sequences are not in A.P., since the common difference is not same in the successive terms. Even though the terms are not in A.P., we can find the next terms in the progression. (i) 42, 56, �....; (ii) 81, 243,��.; (iii) 5x^5, 6x^6, ���.. , are the next terms in the above sequences.