to compute the sum of the first integers without iterating def sum_of_n_ ( )return ( ( ) print(sum_of_n_ ( )we do the same benchmark measurement for sum_of_n_ using five different values for ( and )we get the following resultssum is required seconds sum is required seconds sum is required seconds sum is required seconds sum is required seconds there are two important things to notice about this output firstthe times recorded above are shorter than any of the previous examples secondthey are very consistent no matter what the value of it appears that sum_of_n_ is hardly impacted by the number of integers being added but what does this benchmark really tell usintuitivelywe can see that the iterative solutions seem to be doing more work since some program steps are being repeated this is likely the reason it is taking longer alsothe time required for the iterative solution seems to increase as we increase the value of howeverthere is problem if we ran the same function on different computer or used different programming languagewe would likely get different results it could take even longer to perform sum_of_n_ if the computer were older we need better way to characterize these algorithms with respect to execution time the benchmark technique computes the actual time to execute it does not really provide us with useful measurementbecause it is dependent on particular machineprogramtime of daycompilerand programming language insteadwe would like to have characterization that is independent of the program or computer being used this measure would then be useful for judging the algorithm alone and could be used to compare algorithms across implementations big- notation when trying to characterize an algorithm' efficiency in terms of execution time,independent of any particular program or computerit is important to quantify the number of operations or steps that the algorithm will require if each of these steps is considered to be basic unit of computationthen the execution time for an algorithm can be expressed as the number of steps required to solve the problem deciding on an appropriate basic unit of computation can be complicated problem and will depend on how the algorithm is implemented good basic unit of computation for comparing the summation algorithms shown earlier might be to count the number of assignment statements performed to compute the sum in the function sum_of_nthe number of assignment statements is (the_sum plus the value of (the number of times we perform the_sum=the_sum+iwe can denote this by functioncall it algorithm analysis