Sums of Power

In definite integrals, sum formulas for consecutive integers, squares, and cubes estimate the area using Riemann sums. They simplify calculations over a sequence or pattern.

Consider a section of a large theater, where the first row has one seat, and each new row adds one more seat, forming a sequence up to 35 rows.

The total number of seats is found using the sum formula for the first n consecutive integers, where n is the number of rows. After substituting and solving, the total number of seats can be easily calculated.

A similar concept appears in grocery stores, where cans are stacked in a square pyramid for display. Each layer has a square number of cans—one on top, four on the second layer, nine on the third, and so on.

To find the total number of cans in a five-layer display, the sum of squares formula is used, where n is the number of layers. Substituting the value of n into the formula and simplifying gives the total number of cans used in the pyramid.

If the display follows a cube-based stacking pattern instead, the sum of cubes formula instantly finds the total number.