Index manipulation and summation are fundamental tools used in algorithm analysis. They build a quantitative understanding of the number of iterations or operations performed by algorithms, help derive the time complexity expressions that are used to compare algorithmic efficiency, and optimize algorithm performance. In many algorithms, loops are used to repeat a set of instructions$/$statements multiple times. Index manipulation is even more crucial for analyzing the time complexity of algorithms. Examination of the number of iterations performed by a loop help determine the growth rate of the algorithm's time requirement as the input size increases. The index manipulation allows the expression of the number of iterations in terms of the input size, which is the essential step to derive the algorithm's time complexity, such as O$($n$)$ or O$($n$^2),$ where n represents the input size.Conduct research on the processes of time complexity of some of the common algorithms implemented in C$++$ standard library. Based on your research, provide a report on how index manipulation and summation are used in the algorithm analysis for one or a couple of these algorithms.