Sequence Calculator

nth term of both sequence types.

How the Math Works

An arithmetic sequence increases by adding a constant difference between consecutive terms, calculated as a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference. A geometric sequence multiplies by a constant ratio between terms, given by a_n = a_1 * r^(n-1), where r is the common ratio. These formulas allow precise calculation of any term without listing all previous terms, making them powerful tools for pattern recognition and prediction in structured numerical progressions.

Practical Applications

To find specific values in real-world scenarios, first identify whether your sequence follows addition (arithmetic) or multiplication (geometric). For arithmetic sequences, determine your starting value and common difference—for example, calculating salary increments of $2,000 annually. For geometric sequences, find your initial value and growth factor—such as population doubling every decade. Plug these values into the appropriate formula to quickly compute the nth term without manual enumeration.

Day-to-Day Use

Understanding these sequences helps with financial planning, from calculating compound interest on savings to projecting loan payments. They're useful for analyzing growth patterns in biology, economics, and technology trends. Even everyday tasks like predicting when to water plants based on growth cycles or determining optimal pricing strategies for products benefit from recognizing these fundamental mathematical patterns that govern how things grow and change over time.

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