Prove that:

4^n-3n-1 is divisible by 9

Asked by jgjaigupta1998 | 15th May, 2014, 03:08: PM

Expert Answer:

C o n s i d e r comma space space space space 4 to the power of n minus 3 n minus 1 n equals 1 4 to the power of n minus 3 n minus 1 equals 4 minus 3 minus 1 equals 0 comma space space w h i c h space i s space d i v i s i b l e space b y space 9 n equals k 4 to the power of k minus 3 k minus 1 space comma space l e t s space a s s u m e space t h i s space i s space d i v i s i b l e space b y space 9 space space space space... space left parenthesis i right parenthesis N o w comma space w e space h a v e space t o space p r o v e space i t space f o r space n equals k plus 1 4 to the power of k plus 1 end exponent minus 3 open parentheses k plus 1 close parentheses minus 1 equals 4.4 to the power of k minus 3 k minus 4 equals space 4 to the power of k minus 3 k minus 1 plus space 3.4 to the power of k minus 3 space equals space space open parentheses 4 to the power of k minus 3 k minus 1 close parentheses plus space 3 open parentheses 4 to the power of k minus 1 close parentheses space F i r s t space t e r m space i s space d i v i s i b l e space b y space 9 space left square bracket space f r o m space left parenthesis i right parenthesis right square bracket space a n d space s e c o n d space t e r m space i s space a l s o space d i v i s i b l e space b y space 9 space left parenthesis space w h i c h space c a n space b e space p r o v e d space s i m i l a r space w a y right parenthesis

Answered by Avinash Soni | 15th May, 2014, 05:01: PM

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