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Consider a ball being dropped from a certain height onto a hard surface. After each bounce, it rebounds to a fixed fraction of the height it reached in the previous bounce.

The rebound heights follow a geometric pattern, with each new height being a consistent fraction of the one before.

The fixed value used for multiplication is called the common ratio.

The first term represents the initial rebound height, and each subsequent height is calculated by multiplying the previous term by the common ratio.

This process models exponential decay, as the ball loses energy with each bounce.

Subsequently, the total cumulative distance covered after several bounces is calculated by doubling each rebound height to account for both the ascent and descent, except for the initial drop.

Once each term of the sequence is doubled, partial sums—defined as the sum of a fixed number of terms—can be used to calculate the ball’s total cumulative distance covered at any point.

To find the nth partial sum, write the sum and multiply it by the ratio, shifting all terms forward. Subtracting cancels the middle terms, leaving only the first and last.

Factoring and dividing yield the nth partial sum, showing how geometric sequences work in real situations.