Question

I need help on b-e. THANK YOU

blem 3. Consider the following statement: 1 For all n EN, 12 +22 +32 + ... +n? n(n+1)(2n +1) (a) Prove the statement () using mathematical induction. We use the term closed form expression to describe an algebraic expression that involves only a fixed amount of operations (i.e. that doesn't involving adding n terms). So for example, in the proposition above, the sum of n consecutive natural numbers (12 +22 + ... + n2) has the closed form n(n+1)(2n + 1). (b) Manipulate the result () to find a closed form expression equal to the sum of the first n squared consecutive even whole numbers: 22 +42 +62 + +...+(2n) Note that you do not need to prove (*) via induction, since it relies on the previous result (4) (c) Demonstrate that the closed form expression you came up with in part (b) is accurate for n = 6, that is, it equals the sum 22 +42 +62 +82 +102 + 122. (d) Use your two results (+ and *) to find a closed form expression for the sum of the first n squared consecutive positive odd numbers: 12 + 32 +52 +...+(2n – 1)2 (A) (e) Demonstrate that the closed form expression you found for (A) is accurate for n = 6.