Geometric sequences

Japanese school year: Math B

What you learn

Learn how to find the general term and the sum of geometric sequences. These principles are frequently used to model compound interest, population growth, and other multiplicative processes. Familiarity with exponent rules and basic algebraic operations provides a strong foundation for this topic.

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Key points

Formula to find the $n$ -th term $a_n$ in a geometric sequence with first term $a_1$ and common ratio $r$ . It multiplies $a_1$ by $r$ a total of $n - 1$ times.

a_n = a_1 r^{n-1}

Formula for the sum $S_n$ of the first $n$ terms of a geometric sequence when ratio $r \neq 1$ . When $r = 1$ , the sum is simply $n a_1$ .

S_n = \frac{a_1(1 - r^n)}{1 - r} \quad (r \neq 1)

When three numbers $a, b$ , and $c$ form a geometric sequence in order, the square of the middle term $b^2$ equals the product of the ends $ac$ .

b^2 = ac

Choose a set to practice.

Practice 1Coming soon