need help with this

Problem 1. Consider the following sequence E,, and of partial sums S, , where S,, - E1 + Ez + . .. + En = ) Ek K = For example, since we know E, = 1 and Ez - -4, we calculate S, - E1 + 62 = 1 + (-4) = -3. Register 2 3 4 5 6 Entry (En) 16 (-1)" In2 Sum (S,,) -3 ??? (a) Fill in the missing items in this table, where question marks appear. This means finding the two miss- ing entries, and the row of partial sums Ss, $4,, etc. Based on your findings, find a closed form algebraic expression in terms of n that will give you S,. (b) Prove (by induction) that the pattern you found in part (a) for $,, holds. (You need to have the correct closed form pattern from part (a) to do this, so check with me if you're not sure!) i. Show that the formula you found for S,, holds when n - 1 (that is, that S, - 1), This is the base case. ii. Complete the inductive step: show that if$, - the formula you found in part (a), then S, + 1 equals the formula you found in terms of n + 1. (You may use the fact that E, - (-1)" 'n without proof). (c) Using your answers to part (a) and (b) as well as the guidelines for proofs by induction given above, write a complete proof by induction that $,, is given by the formula you found in part (a).Problem 1. Consider the following sequence E,, and of partial sums S, , where S,, - E1 + Ez + . .. + En = ) Ek K = For example, since we know E, = 1 and Ez - -4, we calculate S, - E1 + 62 = 1 + (-4) = -3. Register 2 3 4 5 6 Entry (En) 16 (-1)" In2 Sum (S,,) -3 ??? (a) Fill in the missing items in this table, where question marks appear. This means finding the two miss- ing entries, and the row of partial sums Ss, $4,, etc. Based on your findings, find a closed form algebraic expression in terms of n that will give you S,. (b) Prove (by induction) that the pattern you found in part (a) for $,, holds. (You need to have the correct closed form pattern from part (a) to do this, so check with me if you're not sure!) i. Show that the formula you found for S,, holds when n - 1 (that is, that S, - 1), This is the base case. ii. Complete the inductive step: show that if$, - the formula you found in part (a), then S, + 1 equals the formula you found in terms of n + 1. (You may use the fact that E, - (-1)" 'n without proof). (c) Using your answers to part (a) and (b) as well as the guidelines for proofs by induction given above, write a complete proof by induction that $,, is given by the formula you found in part (a)