Sequence Calculator

Mathematical sequences follow strict iterative or explicit formulas depending on their classification. For an arithmetic sequence, each term is found by adding a constant common difference ($d$) to the preceding term, expressed explicitly as $a_n = a_0 + n \cdot d$. For a geometric sequence, terms grow multiplicatively via a common ratio ($r$), calculated as $a_n = a_0 \cdot r^n$. Other advanced pattern types—such as Fibonacci numbers, triangular numbers, squares, cubes, and prime numbers—utilize specialized combinatorial and recursive functions to determine successive outputs accurately.

Let us evaluate an arithmetic sequence where the initial term ($a_0$) is 4 and the common difference ($d$) is 3. We want to find the first five sequential values starting from index one. Using the rule $a_n = a_{n-1} + d$:
• For the first term ($n=1$): $4 + (1 \cdot 3) = 7$
• For the second term ($n=2$): $7 + 3 = 10$
• For the third term ($n=3$): $10 + 3 = 13$
• For the fourth term ($n=4$): $13 + 3 = 16$
• For the fifth term ($n=5$): $16 + 3 = 19$
The resulting arithmetic progression is 7, 10, 13, 16, and 19.

When working with numerical series, always double-check whether your pattern relies on addition (linear growth) or multiplication (exponential growth), as confusing the two leads to massive calculation errors. Pay close attention to your starting index (whether counting begins at $n=0$ or $n=1$), as an off-by-one error shifts your entire output. Finally, use the explicit formula method rather than recursive iteration when you need to jump straight to a distant term, such as the 100th element.