Recovering Totals from Successive Changes

From Successive Changes to a Net Change

A quantity may be recorded at a sequence of times or stages. The entries in such a record can show the quantity itself, such as a tank’s volume, or the change in the quantity from one observation to the next. These two kinds of information are linked: adding the successive signed changes recovers the overall change from the first observation to the last.

This link is useful for checking cumulative tables. If a table begins with one value and reports changes at each step, the final value must agree with the sum of those changes. In motion, the same idea says that adding signed displacements over consecutive time intervals gives the net displacement over the whole time period.

The word signed matters. An increase contributes positively and a decrease contributes negatively. The sum of the changes gives the net change, not necessarily the total amount of increase or decrease considered separately.

The Telescoping Sum

Write each change as the value after the step minus the value before it. When these expressions are added, every intermediate value appears once with a positive sign and once with a negative sign. Only the last and first values remain.

This cancellation pattern is called telescoping: the intermediate terms cancel like the overlapping sections of a collapsing telescope. The identity does not require the changes to be equal, or all to have the same sign. It only requires that the change at each step be calculated from the values at the two consecutive endpoints.

The equality can also be read in the opposite direction. If the initial value and all the changes are known, the final value is the initial value plus the sum of the changes. Each change updates the current value, so the updates can be carried out one at a time or checked altogether against the endpoint difference.

Worked Example: Checking a Cumulative Table

A useful audit therefore has two forms. Reconstructing the table checks each individual step; comparing the total of the changes with the endpoint difference checks the overall result. Agreement in the second check does not by itself prove that every intermediate entry is correct, because errors in different rows might offset one another. When the intermediate data matter, check each step as well.

Using the Identity to Find a Missing Change

If one change in a sequence is unknown, the endpoint difference determines what the total of all changes must be. Subtract the known changes from that required total to find the missing one. This works because the unknown change must make the whole sum equal the endpoint difference.

Successive Changes in Motion

For motion along a line, position plays the role of \(Q\). The displacement over an interval is the ending position minus the starting position. Thus, for consecutive time intervals, the signed displacements add to the overall displacement. A negative displacement indicates motion in the negative direction; it is not a negative distance traveled.

When velocity is constant on an interval, the displacement on that interval is velocity multiplied by elapsed time. If the velocity has different constant values on consecutive intervals, calculate each signed displacement separately, then add them. This is an application of successive changes: each interval changes the position from its value at the beginning to its value at the end.

What the Sum Does—and Does Not—Tell You

The telescoping identity compresses a sequence of changes into one overall change. It tells you the difference between the final and initial values, even if the quantity rose and fell many times in between. It does not tell you the largest or smallest intermediate value, nor how much total increase occurred separately from total decrease.

For motion, this distinction is especially important. Net displacement depends on direction and can be zero even when an object has traveled a positive distance. Total distance is found by adding the lengths traveled on each interval; net displacement is found by adding signed displacements, or simply by subtracting the initial position from the final position.

A Common Sign Mistake

A frequent error is to replace a negative change by its positive magnitude when finding net change. In the motion example, writing \(6+6+8=20\) meters as the net displacement is wrong: it has counted the backward displacement as positive. The correct signed calculation is \(6+(-6)+8=8\) meters. The value \(20\) meters is the total distance, not the net displacement.

The reverse mistake is also possible: canceling opposite-direction changes when the question asks for distance. Read the quantity requested before adding. Net change uses signed changes; total distance uses nonnegative lengths.