Write explicit and recursive formulas for arithmetic and geometric sequences; find a specified term; use sigma notation; and apply the arithmetic and finite geometric series sum formulas.

What this topic is asking

The Regents Algebra II exam (the Building Functions, F-BF, and Linear, Quadratic, and Exponential Models, F-LE, clusters) wants you to write explicit and recursive formulas for arithmetic and geometric sequences, find a given term, use sigma notation, and apply the arithmetic and finite geometric series sum formulas. Sequences connect to linear and exponential growth and appear in modeling contexts.

Arithmetic versus geometric

A sequence is arithmetic if each term differs from the previous by a constant common difference $d$, and geometric if each term is the previous one times a constant common ratio $r$.

Arithmetic sequences grow linearly (constant difference) and geometric sequences grow exponentially (constant ratio), mirroring the linear-versus-exponential distinction. A recursive formula always needs a stated first term.

Finding a term

To find a specific term, substitute into the explicit formula. The exponent and the multiplier use $n - 1$, not $n$, because the first term corresponds to $n = 1$ with no change applied yet. For a geometric sequence with $a_1 = 2$ and $r = 4$, the 5th term is $a_5 = 2(4)^{5-1} = 2(4)^4 = 2(256) = 512$.

Sigma notation and series sums

A series is the sum of the terms of a sequence, written compactly with sigma notation: $\sum_{k=1}^{n} a_k$ means add $a_k$ from $k = 1$ to $k = n$. The sum formulas (both on the reference sheet) let you total many terms at once.

Choosing the right tool

A clarifying point that prevents the most frequent errors is to identify arithmetic versus geometric first, then pick the matching formula. A constant difference signals arithmetic (use $a_1 + (n-1)d$ and the $\frac{n}{2}(a_1 + a_n)$ sum); a constant ratio signals geometric (use $a_1 r^{n-1}$ and the $\frac{a_1 - a_1 r^n}{1 - r}$ sum). The arithmetic sum formula also requires the last term $a_n$, so compute that term first if you are not given it. Because both explicit formulas and both sum formulas are printed on the reference sheet, the credit lies in selecting the correct one and substituting accurately, not in memorizing them.

Try this

Q1. Find the 6th term of an arithmetic sequence with $a_1 = 4$ and $d = 5$. [1 credit]

• Cue. $a_6 = 4 + 5(5) = 29$.

Q2. Write the recursive formula for a geometric sequence with first term 3 and ratio 2. [1 credit]

• Cue. $a_1 = 3$, $a_n = 2 a_{n-1}$.