11) Prove using mathematical induction, that 7 2n + 2 3n-3, 3n-1 is divisible by 25, for all n Ꜫ N.

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Please find below the solution to the asked query :

Pn: 72n+23n-3.3n-1 is divisible by 25 .For n=1P1 : 72+23-3.31-149+1=50 , which is divisible by 25 .So , it is true for n=1 .Let us assume that it is true for n=k .So,72k+23k-3.3k-1=25dNow ,For n=k+1pk+1 : 72k+1+23k+1-3.3k+1-1=72k+2+23k.3k=7272k+23k-3.3k-1-23k-3.3k-1+23k.3k=4972k+23k-3.3k-1-49  23k-3.3k-1 +23.23k-3 . 3 .3k-1=49 25 d -49  23k-3.3k-1+24  23k-3.3k-1=49 25 d -25  23k-3.3k-1=25 49d - 23k-3.3k-1 =25p     P=49d - 23k-3.3k-1 saySo , it is divisible by 25 .therefore it is true for n=k+1 .Hence by Principal of mathematical induction ,it is true for all nN .Hence Proved .
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