In each of the following statements, n represents a positive integer. One of the statements is true, the other is false. (I) For all n, the integer 9n + 7 is a multiple of 8. (II) For all n, the integer 9n + 6 is a multiple of 5.   (a) Write down which statement is false and give a counter-example to show that it is false. (b) Prove the other statement by mathematical induction.

4 + 8 + 12 + . . . + 4n = 2n(n + 1).Use mathematical induction to prove that the statement is true for every positive integer n.

in image below

Discrete Math. Base case: Show that the statement is true for the smallest value of n. Inductive step: Assume the statement is true for some integer k, and then show it is true for k + 1. Show step by step how to solve this. Thank you!

Assume that the sum of the digits of a number, n, in base 10, equals to 16. Example: n= 97 is one such number since 9+7=16, and n=4444 is another What are the possible values for the sum of the digits, S, of 2n? Example: If n=97 then 2n= 2*97=194 so S=1+9+4=14. If n=4444 then 2*4444= 8888 and S=4*8= 32. Find all possible values for S, the sum of the digits of 2n, and prove that no other values are possible for S.

Assume that the sum of the digits of a number, n, in base 10, equals to 16. Example: n= 97 is one such number since 9+7=16, and n=4444 is another What are the possible values for the sum of the digits, S, of 2n? Example: If n=97 then 2n= 2*97=194 so S=1+9+4=14. If n=4444 then 2*4444= 8888 and S=4*8= 32. Find all possible values for S, the sum of the digits of 2n, and prove that no other values are possible for S.

Show that 1+5+9+...+ (4n-3) = 2n²- using mathematical induction.

Consider the following statement. There is an integer n such that 2n² - 5n + 2 is prime. To prove the statement it suffices to find a value of n such that (n, 2n² - 5n+ 2) satisfies the property "2n² - 5n + 2 is prime." Show that you can do this by entering appropriate values for n and 2n² - 5n + 2. (n, 2n² - 5n + 2) = 3 X

Use Induction to prove the following.  You must clearly specify the base case and the inductive case. 11n - 6 is a multiple of 5 for all natural numbers n.

Use Induction to prove the following.  You must clearly specify the base case and the inductive case. 11n - 6 is a multiple of 5 for all natural numbers n.