geometric increase in capacity although the proof of proposition relies on the array being doubled each time we expandthe ( amortized bound per operation can be proven for any geometrically increasing progression of array sizes (see section for discussion of geometric progressionswhen choosing the geometric basethere exists tradeoff between run-time efficiency and memory usage with base of ( doubling the array)if the last insertion causes resize eventthe array essentially ends up twice as large as it needs to be if we instead increase the array by only of its current size ( geometric base of )we do not risk wasting as much memory in the endbut there will be more intermediate resize events along the way still it is possible to prove an ( amortized boundusing constant factor greater than the cyber-dollars per operation used in the proof of proposition (see exercise - the key to the performance is that the amount of additional space is proportional to the current size of the array beware of arithmetic progression primitive operations for an append primitive operations for an append to avoid reserving too much space at onceit might be tempting to implement dynamic array with strategy in which constant number of additional cells are reserved each time an array is resized unfortunatelythe overall performance of such strategy is significantly worse at an extremean increase of only one cell causes each append operation to resize the arrayleading to familiar summation and ( overall cost using increases of or at time is slightly betteras portrayed in figure but the overall cost remains quadratic current number of elements current number of elements ( (bfigure running times of series of append operations on dynamic array using arithmetic progression of sizes (aassumes increase of in size of the arraywhile (bassumes increase of