Suppose a Civil Engineer is offered a contract to compose a set of vertical concrete bars around newly built stairs in a way such that the lengths of the concrete bars are given by A=2k-1 where 4 is the length of metal bar of the set. Suppose the civil engineer is interested to find the following finite sums to be carried out the construction as needed by the client. Using the principles of Series and Arithmetic series ((i) to (iv)), (i) Prove that for each metal bar nN, 4 = n, (!!!) (!!) Show that for each metal bar ne N, Use part (i) and part (ii) to deduce that k-1 k-l k="(n+1) 2 (4+2)=n(2+n) for each metal bar neN. k=1 50 30 (iv) Also using the above formulas, calculate A and Ak- (v) k-1 k-20 Find the n" partial sum of the series [ 3(-4)". Is 3(-4)" convergent? Prove n=1 n=1 your answer. Also, is the series 5(3-1)" 5(3-1)" convergent? Prove your answer. n=1