Sequences & series: Algebra 2 Regents Practice

An arithmetic sequence has a first term of $7$ and a common difference of $4$. What is the $15$th term?

• (A) 56
• (B) 63 ✓
• (C) 60
• (D) 67

In an arithmetic sequence each term increases by a fixed common difference, so the $n$th term is $a_n = a_1 + (n-1)d$.

A geometric sequence has a first term of $3$ and a common ratio of $2$. What is the $6$th term?

• (A) 13
• (B) 48
• (C) 96 ✓
• (D) 192

In a geometric sequence each term is multiplied by a fixed common ratio, so the $n$th term is $a_n = a_1 r^{\,n-1}$.

Evaluate the sum $\displaystyle\sum_{k=1}^{5}(3k - 2)$.

• (A) $43$
• (B) $13$
• (C) $45$
• (D) $35$ ✓

Sigma notation directs you to substitute each integer index into the expression and then add all of the resulting terms.

A sequence is defined recursively by $a_1 = -3$ and $a_n = a_{n-1} + 7$. Which explicit formula represents this sequence?

• (A) $a_n = -7n + 4$
• (B) $a_n = 7n - 10$ ✓
• (C) $a_n = 7n + 4$
• (D) $a_n = 7n - 3$

Convert a recursive arithmetic rule to explicit form with $a_n=a_1+(n-1)d$, where $d$ is the constant added at each step.

A sequence is defined recursively by $a_1 = 3$ and $a_n = 2a_{n-1} - 1$ for $n > 1$. What is the value of $a_4$?

• (A) $9$
• (B) $33$
• (C) $15$
• (D) $17$ ✓

A recursive formula builds each term from the previous one, so compute the terms in order: $3, 5, 9, 17$. Stopping one step early gives 9, and going one step too far gives 33.

A geometric sequence is defined by $a_n = 3(2)^{n-1}$. What is the value of $a_5$?

• (A) $48$ ✓
• (B) $24$
• (C) $96$
• (D) $30$

Substitute $n = 5$ into the explicit formula: $a_5 = 3(2)^4 = 3 \cdot 16 = 48$. Using exponent 5 instead of 4 gives 96, and stopping at $a_4$ gives 24.

The first three terms of a geometric sequence are $5,\ 15,\ 45,\ldots$. Write an explicit formula for $a_n$, the $n$th term of the sequence, and use it to find the sum of the first 8 terms.

A geometric sequence has the form $a_n=a_1 r^{n-1}$, and here $a_1=5$ with ratio $r=3$. The finite geometric series formula then gives $S_8=5(6561-1)/2=16400$.

A geometric sequence has first term $a_1 = 3$ and common ratio $r = 2$. Determine the sum of the first 10 terms of this sequence.

The finite geometric series formula $S_n=\frac{a_1(1-r^n)}{1-r}$ sums the terms without listing them all. With $2^{10}=1024$, the computation gives $3(1023)=3069$.

The first three terms of a geometric sequence are $64, 32, 16, \ldots$ Write an explicit formula for $a_n$, the $n$th term of this sequence, and determine the sum of the first six terms of the sequence.

The common ratio is $1/2$, so the explicit formula follows the pattern $a_1 r^{n-1}$. The finite geometric series formula $S_n = a_1(1-r^n)/(1-r)$ then gives $S_6 = 126$, which can be checked by adding $64+32+16+8+4+2$.

What is the sum of the first four terms of the geometric series $3 + 6 + 12 + \dots$ ?

• (A) $24$
• (B) $45$ ✓
• (C) $21$
• (D) $93$

A geometric sequence multiplies by a fixed ratio (here 2). Add the requested number of terms.

What is the sum of the first $20$ terms of the arithmetic sequence with $a_1 = 5$ and common difference $d = 3$?

• (A) 1340
• (B) 62
• (C) 670 ✓
• (D) 700

The sum of the first $n$ terms of an arithmetic sequence is $S_n = \frac{n}{2}(2a_1 + (n-1)d)$.

What is the sum of the first $8$ terms of the geometric series with $a_1 = 2$ and common ratio $r = 3$?

• (A) 4374
• (B) 13120
• (C) 2186
• (D) 6560 ✓

A finite geometric series sums to $S_n = \frac{a_1(r^{\,n} - 1)}{r - 1}$ for $r \neq 1$.

Evaluate $\displaystyle\sum_{k=1}^{5} (2k + 1)$.

• (A) 30
• (B) 25
• (C) 35 ✓
• (D) 31

Sigma notation tells you to substitute each integer index and add the results; here $k$ runs from $1$ to $5$.

A sequence is defined recursively by $a_1 = 4$ and $a_n = 3a_{n-1} - 5$ for $n \ge 2$. What is $a_4$?

• (A) 43 ✓
• (B) 108
• (C) 16
• (D) 103

A recursive formula builds each term from the previous one, so you must compute the terms in order until you reach the one requested.

A sequence is defined recursively by $a_1 = 3$ and $a_n = 2a_{n-1} + 1$ for $n \ge 2$. What is the value of $a_4$?

• (A) $15$
• (B) $30$
• (C) $63$
• (D) $31$ ✓

A recursive rule builds each term from the one before it, so the terms must be generated in order up to the index requested.

Which expression in sigma notation represents the sum $5 + 8 + 11 + 14 + 17$?

• (A) $\displaystyle\sum_{k=1}^{5}(3k + 2)$ ✓
• (B) $\displaystyle\sum_{k=1}^{5}(2k + 3)$
• (C) $\displaystyle\sum_{k=1}^{6}(3k + 2)$
• (D) $\displaystyle\sum_{k=1}^{5}(3k + 5)$

Write the general term using $a_k=a_1+(k-1)d$, simplify it, and set the upper limit equal to the number of terms.

In a geometric sequence, three consecutive terms are $8,\ x,\ 50$. What is the positive value of $x$?

• (A) $29$
• (B) $400$
• (C) $20$ ✓
• (D) $200$

For consecutive geometric terms $a,\,x,\,b$, the middle term satisfies $x^2=ab$, so $x=\sqrt{ab}$ is the geometric mean.

A swimmer covers 800 meters in week 1 of training and increases the distance by 12% each week. Which formula gives the distance $a_n$, in meters, in week $n$?

• (A) $a_n = 800(1.12)^{\,n}$
• (B) $a_n = 800 + 96(n-1)$
• (C) $a_n = 800(1.12)^{\,n-1}$ ✓
• (D) $a_n = 800(0.12)^{\,n-1}$

Repeated percent growth is geometric: multiply the first term by $(1+r)^{\,n-1}$ rather than adding a constant amount each period.

What is the sum of the first 8 terms of the geometric series $3 + 6 + 12 + 24 + \cdots$?

• (A) $765$ ✓
• (B) $1533$
• (C) $384$
• (D) $381$

With first term 3 and common ratio 2, the finite geometric sum formula gives $3(2^8-1)=765$. Distractors are the 8th term itself (384), the sum of only 7 terms (381), and the sum of 9 terms (1533).

The seats in a theater are arranged so that the first row has 22 seats and each row after the first has 4 more seats than the row before it. The theater has 15 rows. Determine the total number of seats in the theater.

The row sizes 22, 26, 30, and so on form an arithmetic sequence with common difference 4. The 15th row has $22+14(4)=78$ seats, and the sum formula $S_n=\tfrac{n}{2}(a_1+a_n)$ gives $\tfrac{15}{2}(22+78)=750$ total seats.

Alejandro accepts a job with a starting salary of $\$42{,}000$ for his first year, and he will receive a $2.5\%$ raise at the end of each year. Write a formula for $a_n$, his salary in year $n$. Then determine, to the nearest dollar, the total amount Alejandro will earn over his first 15 years at the job.

A fixed percent raise makes the yearly salaries a geometric sequence with common ratio $1.025$. The total earned over 15 years is the sum of the first 15 terms, computed with the finite geometric series formula $S_n = a_1(1 - r^n)/(1 - r)$, giving about $753,141 dollars.

What is the sum of the first eight terms of the geometric series $3+6+12+24+\cdots$?

• (A) $384$
• (B) $381$
• (C) $765$ ✓
• (D) $768$

The series is geometric with first term 3 and common ratio 2. Applying $S_n=\frac{a_1(1-r^n)}{1-r}$ with $n=8$ gives $3(255)=765$. Using only seven terms gives 381, and $384$ is just the eighth term itself, $3\cdot 2^7$.

A sequence is defined recursively by $a_1=3$ and $a_{n+1}=2a_n-1$. What is the value of $a_4$?

• (A) $24$
• (B) $9$
• (C) $33$
• (D) $17$ ✓

Apply the recursive rule one term at a time: $a_2=5$, $a_3=9$, and $a_4=17$. Stopping one step too early gives $a_3=9$, and iterating one extra time gives $a_5=33$, both common miscounting errors with recursive definitions.

What is the value of $\displaystyle\sum_{k=1}^{6} 3(2)^{k-1}$?

• (A) $96$
• (B) $381$
• (C) $93$
• (D) $189$ ✓

The summation is a finite geometric series with first term 3, common ratio 2, and 6 terms. Applying $S_n = \frac{a_1(1-r^n)}{1-r}$ gives $\frac{3(1-64)}{-1} = 189$, which matches adding the terms directly. Using only 5 terms gives the distractor 93, and 96 is just the last term.