Task Sets: Write arithmetic and geometric sequences - Common Core Math - HS Functions

A

Warm-Up: Review What You Know

1.

The sequence 4, 12, 36, 108, ... is best classified as:

A.

Arithmetic — because the terms are increasing

B.

Geometric — because consecutive terms have a constant ratio

C.

Arithmetic — because the differences 8, 24, 72 are increasing by a factor of 3

D.

Neither arithmetic nor geometric

2.

For the arithmetic sequence 7, 11, 15, 19, ..., what is the common difference $d$ ?

3.

For the geometric sequence 5, 10, 20, 40, ..., what is the common ratio $r$ ?

B

Fluency Practice

1.

An arithmetic sequence has $a_1 = 3$ and $d = 5$ . Using the explicit formula $a_n = a_1 + (n-1)d$ , find $a_{10}$ .

2.

A job pays $32,000 in year 1 with a $2,500 raise each year. Using the arithmetic explicit formula, what is the salary in year 8?

3.

A geometric sequence has $a_1 = 2$ and $r = 3$ . Using the explicit formula $a_n = a_1 \cdot r^{n-1}$ , find $a_5$ .

4.

A car worth $24,000 depreciates 20% per year. Which explicit formula gives its value after $n$ years?

A.

$a_n = 24000 \cdot (0.2)^{n-1}$

B.

$a_n = 24000 - 20n$

C.

$a_n = 24000 \cdot (0.8)^{n-1}$

D.

$a_n = 24000 \cdot (0.8)^n$

5.

The explicit formula $a_n = 7 + (n-1) \cdot 4$ corresponds to a recursive definition with $a_1 = \text{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}}$ and $a_n = a_{n-1} + \text{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}}$ .

initial value a₁:

common difference d:

C

Varied Practice

1.

For the arithmetic sequence 10, 7, 4, 1, ..., the explicit formula is $a_n = 10 + (n-1) \cdot (\text{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}})$ , and $a_6 = \text{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}}$ .

common difference d:

sixth term:

2.

A geometric sequence starts at $a_1 = 100$ with common ratio $r = 0.5$ . The recursive definition is $a_1 = 100$ and $a_n = a_{n-1} \cdot \text{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}}$ . The explicit formula is $a_n = 100 \cdot (\text{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}})^{n-1}$ .

common ratio (in recursion):

common ratio (in explicit):

3.

A sequence of dot patterns is shown: row 1 has 3 dots, row 2 has 5 dots, row 3 has 7 dots. Which explicit formula gives the number of dots in row $n$ ?

A.

$a_n = 3n$

B.

$a_n = 2n + 1$

C.

$a_n = 3 \cdot 2^{n-1}$

D.

$a_n = 2n + 3$

4.

The sequence 1, -2, 4, -8, 16, ... is geometric with a negative ratio. What is the common ratio $r$ ?

5.

The sequence 1, 3, 6, 10, 15, ... has differences of 2, 3, 4, 5, ... and ratios of 3, 2, 5/3, 3/2, ... Is it arithmetic, geometric, or neither? Explain.

D

Word Problems

1.

A teacher's salary starts at $40,000 and increases by $1,500 each year.

1.

Write the explicit formula for the salary in year $n$ and find the salary in year 15.

2.

Using the recursive definition ( $a_1 = 40000$ , $a_n = a_{n-1} + 1500$ ), what is the salary in year 3?

2.

A car costs $30,000 and depreciates by 15% each year (worth 85% of the previous year's value).

Using the geometric explicit formula, find the car's value after 5 years. Round to the nearest dollar.

3.

A tree grows 2 feet per year starting at 5 feet tall. A bacteria culture doubles every hour starting with 500 bacteria.

How should these two situations be modeled?

A.

Both are arithmetic — both involve growth

B.

Both are geometric — both are increasing

C.

Tree: arithmetic (constant addition); Bacteria: geometric (constant multiplication)

D.

Tree: geometric (growing); Bacteria: arithmetic (predictable)

E

Error Analysis

1.

For the sequence 2, 6, 18, 54, ..., a student writes: "This is arithmetic with $d = 4$ . The explicit formula is $a_n = 2 + (n-1)(4)$ ."

What is the student's error?

A.

The student computed the differences but should have computed ratios — the sequence is geometric

B.

The common difference is wrong — it should be 6, not 4

C.

The explicit formula structure is wrong for an arithmetic sequence

D.

This sequence is neither arithmetic nor geometric

2.

A student writes the explicit formula for the arithmetic sequence $a_1 = 5$ , $d = 3$ : " $a_n = 5 + 3n$ ."

What is wrong with this formula?

A.

The formula uses $n$ instead of $(n-1)$ , making the first term wrong

B.

The formula should use $n-1$ for the first term: $a_n = (5-1) + 3n$

C.

Nothing is wrong — $a_n = 5 + 3n$ is equivalent to $a_n = 5 + 3(n-1)$

D.

The common difference should be n, not 3

F

Challenge / Extension

1.

The recursive definition $a_1 = 4$ , $a_n = a_{n-1} \cdot 0.5$ converts to the explicit formula $a_n = 4 \cdot (\text{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}})^{n-1}$ . Then $a_7 = \text{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}}$ .

common ratio:

seventh term:

2.