Tiêu đề XI - maths - chapter 4 - MATHEMATICAL INDUCTION (68-83).pdf ✅

MATHEMATICAL INDUCTION 68 NARAYANAGROUP JEE-MAIN-JR-MATHS VOL-I  Principle of Finite Mathematical Induction: For n N  , let P(n) be a statement in terms of n. If P(1) is true and P(k) is true  P(k + 1) is true, then P(n) is true, for all n N  .  Principle of Complete Mathematical Induction: For n N  , let P(n) be a statement in terms of n. If P(1), P(2), P(3),.... P(k-1) are true  P(k) is true, then P(n) is true, for all n N  .  a a d a d , , 2 , ...       form an A.P. then (i) nth term t a n d n     1 , Where a is the first term and d is the common difference. (ii) Sum of n terms 2 1   2 n n S a n d          2 n   a l Where a = first term, l = last term  2 , a a r a r , , ... form a G.P then (i) nth term 1 . n nt a r   , Where a = first term r = common ratio (ii) Sum of n terms  1; 1 n n r S a r    (iii) In an infinite G.P, Sum of Infinite terms is 1 a S r    where r 1      2 a a d r a d r      2 ...   1 ...... 1 n a n d r         form A.G.P. then (i) nth term   1 1 n nt a n d r         (ii) Sum of n terms         1 2 1 1 1 1 1 n n n a dr r a n d r S r r r               (iii) Sum of Infinite terms   2 1 1 1 a dr S where r r r       W.E-1: Sum to infinite terms of 2 3 1 3 5 7 .....       x x x (If x 1)is__ [Eam-1998] Sol:   2 1 1 a dr s r r      where a 1, d  2 ,r x      2 2 1 2 1 1 1 1 x x x x x        Similarly If x 1,   2 3 2 1 1 2 3 4 ... 1 s x x x x           SOME IMPORTANT POINTS i) Sum of first n natural numbers i.e. n n 1   n 1 2 3 ......n 2        , n N ii) Sum of the squares of first n natural numbers is    2 2 2 2 2 1 2 1 1 2 3 ... , 6 n n n n n n N            iii) Sum of the cubes of first n natural numbers is   2 2 3 3 3 3 3 1 1 2 3 ... 4 n n n n           2    n n N , iv) 4 4 4 4 1 2 3 ......      n 4 n      4 2 1 1 3 1 5 6 n n n n n n       v) Sum of the first ‘n’ odd +ve integers = 1 + 3 + 5 + ... + (2n-1) 2  n vi) Sum of the first ‘n’ even +ve integers = 2 + 4 + 6 + ............ + 2n   n n 1 MATHEMATICAL INDUCTION SYNOPSIS