Please solve ques 7 and 9 Thank you - Maths - Principle of Mathematical Induction

Question

Please solve ques 7 and 9
Thank you

Ram Manohar Thakur · Accepted Answer

Dear Student ,
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 n∈N 
Hence Proved

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