CBSE Class 12-science Answered

Asked by vatsala srivastava srivastava | 16 Jun, 2014, 09:57: PM

Expert Answer

C o n s i d e r space t h e space g i v e n space e q u a t i o n space A squared equals 2 A minus I.... left parenthesis 1 right parenthesis W e space n e e d space t o space p r o v e space t h a t space f o r space n greater or equal than 2 comma space A to the power of n equals n A minus open parentheses n minus 1 close parentheses I. L e t space u s space p r o v e space t h i s space b y space t h e space m e t h o d space o f space i n d u c t i o n. C o n s i d e r space A to the power of n equals n A minus open parentheses n minus 1 close parentheses I. W h e n space n equals 2 comma space w e space h a v e comma A squared equals 2 A minus open parentheses 2 minus 1 close parentheses I rightwards double arrow space A squared equals 2 A minus I comma space w h i c h space i s space t r u e. L e t space u s space a s s u m e space t h a t space A to the power of n equals n A minus open parentheses n minus 1 close parentheses I space i s space t r u e space f o r space n equals k comma space 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 t h i s space i s space t r u e space f o r n equals k plus 1 S i n c e space t h e space r e q u i r e d space s t a t e m e n t space i s space t r u e space f o r space n equals k comma space w e space h a v e comma A to the power of k equals k A minus open parentheses k minus 1 close parentheses I. W e space n e e d space t o space p r o v e space t h a t space A to the power of k plus 1 end exponent equals left parenthesis k plus 1 right parenthesis A minus k I. C o n s i d e r space t h e space s t a t e m e n t A to the power of k equals k A minus open parentheses k minus 1 close parentheses I M u l t i p l y space b o t h space t h e space s i d e s space o f space t h e space e q u a t i o n space b y space A comma space w e space h a v e A to the power of k cross times A equals open square brackets k A minus open parentheses k minus 1 close parentheses I close square brackets cross times A rightwards double arrow A to the power of k plus 1 end exponent equals k A squared minus open parentheses k minus 1 close parentheses A rightwards double arrow A to the power of k plus 1 end exponent equals k open parentheses 2 A minus I close parentheses minus open parentheses k minus 1 close parentheses A..... open square brackets f r o m space e q u a t i o n space left parenthesis 1 right parenthesis close square brackets rightwards double arrow A to the power of k plus 1 end exponent equals 2 A k minus k I minus k A plus A rightwards double arrow A to the power of k plus 1 end exponent equals A k minus k I plus A rightwards double arrow A to the power of k plus 1 end exponent equals A k plus A minus k I rightwards double arrow A to the power of k plus 1 end exponent equals open parentheses k plus 1 close parentheses A minus k I H e n c e space p r o v e d.

Answered by Vimala Ramamurthy | 17 Jun, 2014, 11:04: AM

CBSE Class 12-science Answered

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CBSE Class 12-science Answered

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