Question

Consider the following sum (which is in expanded form): 1−4 + 7−10 + 13−16 + 19−22 +···±(3n−2).

Note that this is slightly different from the previous sum in that every other term is negative.

(a) Write it as a summation (∑).

(b) Evaluate the sum for every integer n from 1 to 9. (Be careful - if you get this wrong, you will likely get the rest of this question wrong!)

(c) Write a closed-form formula for the value of the sum as a function of n. As in problem 1, do not use a "by cases" or piecewise definition (will need to write a single closed-form expression to receive full credit).(Hint 1: floor and ceiling functions may be useful here.)(Hint 2: try splitting up the sequence of partial sums into two subsequences, finding formulas foreach of the subsequences, then combining the formulas.)

(d) Prove that your formula from part (c) is correct using Mathematical Induction. (You may separateout the cases wherenis even/odd if you wish, but if so please do it as late as possible.)

i. State and prove the Base Case.

ii. State the Inductive Hypothesis.

iii. Show the Inductive Step

Answer 1

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