HOW IS THE STEP P(K+1) IS PROVED AFTER THE STEP P(K)?EXPLAIN ME THAT.FOR EXAMPLE K(K+3)DIVIDE BY 4 +  1+2+...+K+1 DIVIDE BY
K+1

Asked by Ishita Roy | 20th Jul, 2014, 09:50: AM

Expert Answer:

M a t h e m a t i c a l space i n d u c t i o n i s space a space m e t h o d space o f space m a t h e m a t i c a l space p r o o f space u s e d space t o space e s t a b l i s h space a space g i v e n space s t a t e m e n t space f o r space a l l space n a t r u a l n u m b e r s. T h e space p r i n c i p l e space o f space m a t h e m a t i c a l space i n d u c t i o n space c o n s i s t s space o f space t w o space s t e p s comma space b a s e space c a s e space a n d space t h e space i n d u c t i v e space c a s e. T h e space b a s e space c a s e space p r o v e s space t h a t space t h e space s t a t e m n e t space h o l d s space f o r space t h e space f i r s t space n a t u r a l space n u m b e r comma space t h a t space i s space k equals 1 T h e space i n d u c t i v e space s t e p space p r o v e s space t h a t space i f space t h e space s t a t e m e n t space h o l d s space f o r space k comma space t h e n space t h e space s t a t e m e n t space h o l d s space f o r space k plus 1. E x a m p l e : P r o v e space t h a t space k open parentheses k plus 3 close parentheses space i s space d i v i s i b l e space b y space 2  P r o o f : S t e p 1 : space P u t space k equals 1 S i n c e space 1 cross times open parentheses 1 plus 3 close parentheses equals 4 comma space i t space i s space p r o v e d space t h a t space t h e space g i v e n space s t a t e m e n t space i s space t r u e space f o r space k equals 1 S t e p 2 : space N o w space a s s u m e space t h a t space t h e space s t a t e m e n t space i s space t r u space f o r space k. A n d space w e space n e e d space t o space p r o v e space t h a t space i t space i s space t r u e space f o r space k plus 1 S i n c e space t h e space s t a t e m e n t space i s space t r u e space f o r space k comma space w e space h a v e comma P open parentheses k close parentheses equals k cross times open parentheses k plus 3 close parentheses space i s space d i v i s i b l e space b y space 2 rightwards double arrow k open parentheses k plus 3 close parentheses equals 2 lambda comma space f o r space lambda element of N N o w space c o n s i d e r open parentheses k plus 1 close parentheses open parentheses k plus 1 plus 3 close parentheses equals open parentheses k plus 1 close parentheses open parentheses k plus 4 close parentheses equals k squared plus 5 k plus 4 equals k squared plus 3 k plus 2 k plus 4 equals k squared plus 3 k plus 2 k plus 4 equals 2 lambda plus 2 open parentheses k plus 2 close parentheses equals 2 open parentheses lambda plus k plus 2 close parentheses T h u s space i t space i s space p r o v e d space t h a t space space k open parentheses k plus 3 close parentheses space i s space d i v i s i b l e space b y space 2. P l e a s e space p o s t space a space f r e s h space q u e r y comma space i f space t h e space q u e s t i o n space a n d space s o l u t i o n space d e s c r i b e d space a b o v e space i s space n o t space t h e space r e u i r e d space o n e.

Answered by Vimala Ramamurthy | 21st Jul, 2014, 12:13: PM