 Hello friends, let's discuss the following question. It says show that 9 to the power n plus 1 minus 8n minus 9 is divisible by 64 whenever n is a positive integer. Now to solve this question we need to know that m is divisible by n then m is equal to some k into n where k is some natural number. We also need to know the expansion of a plus b whole to the power n it is given by nc0 a to the power n plus nc1 a to the power n minus 1 b plus so on the last term is ncn b to the power n. So this knowledge will work as k idea. Now we have to show 9 to the power n plus 1 minus 8n minus 9 is divisible by 64 that is we have to show 9 to the power n plus 1 minus 8n minus 9 is equal to 64 into k where is some natural number. Let's now start the solution 9 can be written as 1 plus 8 so 9 to the power n plus 1 can be written as 1 plus 8 whole to the power n plus 1. Now we will expand this using this expansion so 1 plus 8 whole to the power n plus 1 is equal to n plus 1 c0 1 to the power n plus 1 plus n plus 1 c1 1 to the power n plus 1 minus 1 into 8 plus n plus 1 c2 1 to the power n plus 1 minus 2 into 8 to the power 2 plus n plus 1 c3 1 to the power n plus 1 minus 3 into 8 to the power 3 so on the last term is n plus 1 cn plus 1 8 to the power n. Now this is equal to the first term is 1 because n plus 1 c0 is 1 and 1 to the power n plus 1 is 1. The second term is 8 into n plus 1 because n plus 1 c1 is n plus 1 and 1 to the power n is 1 so on 8 square into n plus 1 c2 plus 8 cube n plus 1 c3 so on the last term is 8 to the power n n plus 1 cn plus 1. Now this is equal to 1 plus 8 into n is 8n 8 into 1 is 8 plus 8 square into n plus 1 c2 plus 8 cube n plus 1 c3 so on last term is 8 to the power n because n plus 1 cn plus 1 is 1. Now this is equal to 8n plus 9 taking 8 square common from all these terms we get 8 square into n plus 1 c2 plus 8 into n plus 1 c3 so on last term is 8 to the power n minus 2. Now again this is equal to 8n plus 9 plus 64 into n plus 1 c2 plus 8 into n plus 1 c3 so on 8 to the power n minus 2 which is again equal to 8n plus 9 plus 64 into k where equal to n plus 1 c2 plus 8 into n plus 1 c3 so on last term is 8 to the power n minus 2 so we have proved that 9 to the power n plus 1 is equal to 8n plus 9 plus 64 k this implies 9 to the power n plus 1 minus 8n minus 9 is equal to 64 k and this implies 9 to the power n plus 1 minus 8n minus 9 is divisible by 64 the result is proved. This concludes the question hope you enjoy the session goodbye and take care.