What is the general term formula for a geometric series?

What is the general term formula for a geometric series?

• A. An = a1 * n^(r-1)
• B. An = a1 * r^(n-1)
• C. An = a1 + n
• D. An = a1 * (n/r)

Answer: (B)

The correct formula for the general term of a geometric series is given by the expression An = a1 * r^(n-1). This formula allows you to find any term in a geometric sequence where "a1" represents the first term of the series, "r" is the common ratio (the factor by which each term is multiplied to get the next term), and "n" is the term number you want to find. In a geometric series, each term is derived by multiplying the previous term by the common ratio. Therefore, to derive the nth term, you start from the first term, a1, and continually apply the common ratio, r, for each subsequent term. The exponent (n-1) reflects how many times you multiply by the common ratio since the first term does not require a multiplication. This principle illustrates the nature of geometric growth, where the terms can grow or shrink rapidly based on the value of the common ratio. In summary, this formula captures the essence of geometric sequences, making it an essential concept in understanding the behavior of such series.

The correct formula for the general term of a geometric series is given by the expression An = a1 * r^(n-1). This formula allows you to find any term in a geometric sequence where "a1" represents the first term of the series, "r" is the common ratio (the factor by which each term is multiplied to get the next term), and "n" is the term number you want to find.

In a geometric series, each term is derived by multiplying the previous term by the common ratio. Therefore, to derive the nth term, you start from the first term, a1, and continually apply the common ratio, r, for each subsequent term. The exponent (n-1) reflects how many times you multiply by the common ratio since the first term does not require a multiplication.

This principle illustrates the nature of geometric growth, where the terms can grow or shrink rapidly based on the value of the common ratio. In summary, this formula captures the essence of geometric sequences, making it an essential concept in understanding the behavior of such series.