Write and evaluate explicit and recursive formulas for arithmetic and geometric sequences, and relate arithmetic sequences to linear functions and geometric sequences to exponential functions (MA.912.AR.5 and MA.912.F).

What this topic is asking

A sequence is an ordered list of numbers. On the B.E.S.T. Algebra 1 EOC you identify whether a sequence is arithmetic (constant difference) or geometric (constant ratio), write its explicit and recursive formula, and connect it to a function: arithmetic sequences are linear and geometric sequences are exponential. Both $n$th-term formulas are on the reference sheet, so the credit is for using them correctly.

Telling them apart

Check consecutive terms. If you add the same number each time, it is arithmetic (find $d$ by subtracting any term from the next). If you multiply by the same number each time, it is geometric (find $r$ by dividing any term by the previous one).

• $3, 7, 11, 15, \dots$ is arithmetic with $d = 4$.
• $3, 6, 12, 24, \dots$ is geometric with $r = 2$.

The explicit formulas (on the reference sheet)

The explicit formula gives any term directly from $n$, without listing the ones before it. The key detail is the $n - 1$: the first term has had the difference or ratio applied zero times, so the multiplier or exponent is one less than the term number.

The recursive formulas

A recursive formula defines each term from the previous one and requires a starting value:

• Arithmetic: $a_1 =$ (given), $a_n = a_{n-1} + d$.
• Geometric: $a_1 =$ (given), $a_n = a_{n-1} \cdot r$.

Recursive form is handy for generating the next term quickly, but to jump straight to the 50th term you want the explicit formula.

How the B.E.S.T. EOC examines this topic

• Equation editor. Write the explicit or recursive formula for a given sequence.
• Multiple choice. Find a specific term, with off-by-one ($n$ vs $n - 1$) distractors.
• Editing task choice or matching. Match a sequence to its formula or to linear vs exponential.

A clarifying idea: the simplified explicit arithmetic formula $a_n = 3n + 2$ above is a linear function of $n$ (slope $3$, intercept $2$). That is the deep link the standard wants, an arithmetic sequence is just a line evaluated at the whole numbers.

Why arithmetic is linear and geometric is exponential

The connection to functions is exact, not loose. In an arithmetic sequence you add $d$ each step, so the term grows by the same amount per unit increase in $n$, which is precisely the definition of constant slope, a linear function $a_n = a_1 + d(n - 1)$. In a geometric sequence you multiply by $r$ each step, so the term is scaled by the same factor per unit increase in $n$, which is the defining behavior of an exponential function $a_n = a_1 \cdot r^{n-1}$. This is why the EOC can ask the same idea two ways: a table that grows by adding is linear, a table that grows by multiplying is exponential, and the sequence formulas are those two functions restricted to the counting numbers $1, 2, 3, \dots$.

Try this

Q1. Find $a_8$ for the geometric sequence with $a_1 = 2$ and $r = 3$. [2 points]

• Cue. $a_8 = 2 \cdot 3^{7} = 2 \cdot 2187 = 4374$.

Q2. Write the recursive formula for $7, 4, 1, -2, \dots$. [1 point]

• Cue. $a_1 = 7$, $a_n = a_{n-1} - 3$.