Number and Algebra - Geometric sequences

In a geometric sequence, the first term is $3$ and the second term is $6$. Find the $10^{th}$ term.

$u_1 = 3$ $u_2 = 6$ $r = \frac{6}{3} = 2$ $u_{10} = 3 \times 2^{10-1}$ $u_{10} = 3 \times 2^9$ $u_{10} = 3 \times 512 = 1536$

First, identify the common ratio $r$ by dividing the second term by the first. Then, apply the general term formula $u_n = u_1 r^{n-1}$ for $n=10$.

Find the sum of the first $8$ terms of the geometric sequence $10, 5, 2.5, \dots$

$u_1 = 10$ $r = \frac{5}{10} = 0.5$ $S_8 = \frac{10(1 - 0.5^8)}{1 - 0.5}$ $S_8 = \frac{10(1 - 0.00390625)}{0.5}$ $S_8 = 20 \times 0.99609375 = 19.921875$

Identify $u_1$ and $r$. Since $r = 0.5$, we use the sum formula $S_n = \frac{u_1(1-r^n)}{1-r}$ to calculate the sum of the first 8 terms.

An infinite geometric series has a first term of $12$ and a common ratio of $\frac{1}{3}$. Calculate the sum to infinity.

$u_1 = 12$ $r = \frac{1}{3}$ Since $|\frac{1}{3}| < 1$, the sum to infinity exists: $S_{\infty} = \frac{12}{1 - \frac{1}{3}}$ $S_{\infty} = \frac{12}{\frac{2}{3}}$ $S_{\infty} = 12 \times \frac{3}{2} = 18$

The sum to infinity formula $S_{\infty} = \frac{u_1}{1-r}$ is used because the common ratio satisfies the condition $|r| < 1$.