Use the Principle of Mathematical Induction to show that the following statement is true for all natural numbers n. 3+2+1+ ...+(4-n)= =n(7 -n) ..... What two conditions must the given statement satisfy to prove that it is true for all natural numbers? Select all that apply. The statement is true for the natural number 1. If the statement is true for the natural number 1, it is also true for the next natural number 2. If the statement is true for some natural number k, it is also true for the next natural number k + 1. The statement is true for any two natural numbers k and k + 1. Show that the first of these conditions is satisfied by evaluating the left and right sides of the given statement for the first natural number. 3+2+1+ ...+(4 -n) = n(7-n) =(Simplify your answers.) To show that the second condition is satisfied, write the given statement for k + 1. 3+ 2 + 1+ ... + (4 -k) + =[ (Use integers or fractions for any numbers in the expression. Type your answer in factored form.) Now, according to the Principle of Mathematical Induction, assume that 3 + 2 + 1 + ... +(4-k) = (Simplify your answer. Use integers or fractions for any numbers in the expression. ) Use this assumption to rewrite the left side of the statement for k + 1. What is the resulting expression? Do not simplify. Type your answer in factored form. Use integers or fractions for any numbers in the expression.) What can be done to show that the resulting statement for k + 1 is true?Use the Principle of Mathematical Induction to show that the following statement is true for all natural numbers n. 3+2+1+ ...+(4-n)= =n(7-n) The statement is true for any two natural numbers k and k + 1. Show that the first of these conditions is satisfied by evaluating the left and right sides of the given statement for the first natural number. 3+2+1+...+(4-n) =2n(7-n) 0=(Simplify your answers.) To show that the second condition is satisfied, write the given statement for k + 1. 3+2+1 + ...+ (4-k) +=[ (Use integers or fractions for any numbers in the expression. Type your answer in factored form.) K Now, according to the Principle of Mathematical Induction, assume that 3 + 2+1 + ...+(4-k) = (Simplify your answer. Use integers or fractions for any numbers in the expression. ) Use this assumption to rewrite the left side of the statement for k + 1. What is the resulting expression? Do not simplify. Type your answer in factored form. Use integers or fractions for any numbers in the expression.) What can be done to show that the resulting statement for k + 1 is true? A. Write the terms of the expression on the left side over a common denominator and factor the numerator to make the expression on the left side of the statement identical to the one on the right. O B. Factor k out of each side of the statement to demonstrate that the resulting equation is true for all natural numbers k > 0. O C. Substitute 1 for k and evaluate each side of the statement to show that they have the same value. O D. Multiply the left and right sides of the statement by 2 and simplify each side to the same expression. Since both conditions of the Principle of Mathematical Induction are satisfied, the given statement 3 + 2+ 1+ ... + (4 -n) = ,n(7 -n) is true for all natural numbers