Comprehensive Guide to Arithmetic and Geometric Sequences

An arithmetic sequence is defined as a sequence with a common difference between any two consecutive terms. The first term is denoted as $a<em>1$, and the common difference as $d$. For example, in the sequence $79, 75, 71, 67, 63, ext{…}$, $a</em>1 = 79$ and $d = -4$. To find a specific term, such as the $32^{nd}$ term ($a<em>{32}$), we use the formula $a</em>n = a<em>1 + (n - 1)d$. This results in $a</em>{32} = 79 + (32 - 1)(-4) = -45$. We can also determine the position of a known value. For instance, to find which term equals $-169$, we set the formula equal to $-169$ and simplify to find $n = 63$. In another example, if the first term is $5$ and the common difference is $4$, the $5^{th}$ term is calculated using $a<em>5 = 5 + (5 - 1) imes 4 = 21$. For a geometric sequence, defined by a common ratio, the same principles apply. The general formula here is $a</em>n = a<em>1 imes r^{n-1}$. For instance, to find the $15^{th}$ term of a sequence where $a</em>1 = 20$ and $r = 1.05$, we calculate $a_{15} = 20 imes (1.05)^{14} = 39.599$. Understanding these sequences aids in recognizing patterns and rates of change in various contexts.