Math / Algebra

QuestionThe first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary). $2,7,12, \ldots$
Find the 46 th term.

Studdy Solution

STEP 1

What is this asking? We're figuring out what the 46th number is in a list of numbers that starts with 2, 7, and 12. Watch out! Don't just add 5 to 12 forty-three times!
There's a slicker way.

STEP 2

1. Find the Pattern
2. Create the Formula
3. Calculate the 46th Term

STEP 3

Alright, let's look closely at these numbers: 2, 7, and 12.
What's happening here?

STEP 4

From 2 to 7, we're adding 5.
From 7 to 12, we're also adding 5!
So, this looks like an arithmetic sequence with a common difference of $5$ .

STEP 5

The formula for the nth term of an arithmetic sequence is $a_n = a_1 + (n-1) \cdot d$ , where $a_n$ is the term we're looking for, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

STEP 6

In our case, the first term $a_1$ is $2$ , and the common difference $d$ is $5$ .
We want to find the 46th term, so $n = 46$ .
Let's plug these values into our formula!

STEP 7

Substituting our values, we get $a_{46} = 2 + (46-1) \cdot 5$ .

STEP 8

First, inside the parentheses, $46 - 1 = 45$ .
So, our formula becomes $a_{46} = 2 + 45 \cdot 5$ .

STEP 9

Next, we multiply: $45 \cdot 5 = 225$ .
Now, we have $a_{46} = 2 + 225$ .

STEP 10

Finally, we add: $2 + 225 = 227$ .
So, the 46th term is 227!

STEP 11

The 46th term of the sequence is $\boxed{227}$ .

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QuestionThe first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary). $2,7,12, \ldots$
Find the 46 th term.

Studdy Solution

STEP 1

What is this asking? We're figuring out what the 46th number is in a list of numbers that starts with 2, 7, and 12. Watch out! Don't just add 5 to 12 forty-three times!
There's a slicker way.

STEP 2

1. Find the Pattern
2. Create the Formula
3. Calculate the 46th Term

STEP 3

Alright, let's look closely at these numbers: 2, 7, and 12.
What's happening here?

STEP 4

From 2 to 7, we're adding 5.
From 7 to 12, we're also adding 5!
So, this looks like an arithmetic sequence with a common difference of $5$ .

STEP 5

The formula for the nth term of an arithmetic sequence is $a_n = a_1 + (n-1) \cdot d$ , where $a_n$ is the term we're looking for, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

STEP 6

In our case, the first term $a_1$ is $2$ , and the common difference $d$ is $5$ .
We want to find the 46th term, so $n = 46$ .
Let's plug these values into our formula!

STEP 7

Substituting our values, we get $a_{46} = 2 + (46-1) \cdot 5$ .

STEP 8

First, inside the parentheses, $46 - 1 = 45$ .
So, our formula becomes $a_{46} = 2 + 45 \cdot 5$ .

STEP 9

Next, we multiply: $45 \cdot 5 = 225$ .
Now, we have $a_{46} = 2 + 225$ .

STEP 10

Finally, we add: $2 + 225 = 227$ .
So, the 46th term is 227!

STEP 11

The 46th term of the sequence is $\boxed{227}$ .

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QuestionThe first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary). 2,7,12,…2,7,12, \ldots2,7,12,…Find the 46 th term.

Studdy Solution

STEP 1

What is this asking? We're figuring out what the 46th number is in a list of numbers that starts with 2, 7, and 12. Watch out! Don't just add 5 to 12 forty-three times!There's a slicker way.

STEP 2

1. Find the Pattern2. Create the Formula3. Calculate the 46th Term

STEP 3

Alright, let's look closely at these numbers: 2, 7, and 12.What's happening here?

STEP 4

From 2 to 7, we're adding 5.From 7 to 12, we're also adding 5!So, this looks like an **arithmetic sequence** with a **common difference** of 555.

STEP 5

The formula for the nth term of an arithmetic sequence is an=a1+(n−1)⋅da_n = a_1 + (n-1) \cdot dan​=a1​+(n−1)⋅d, where ana_nan​ is the term we're looking for, a1a_1a1​ is the **first term**, nnn is the term number, and ddd is the **common difference**.

STEP 6

In our case, the **first term** a1a_1a1​ is 222, and the **common difference** ddd is 555.We want to find the 46th term, so n=46n = 46n=46.Let's plug these values into our formula!

STEP 7

Substituting our values, we get a46=2+(46−1)⋅5a_{46} = 2 + (46-1) \cdot 5a46​=2+(46−1)⋅5.

STEP 8

First, inside the parentheses, 46−1=4546 - 1 = 4546−1=45.So, our formula becomes a46=2+45⋅5a_{46} = 2 + 45 \cdot 5a46​=2+45⋅5.

STEP 9

Next, we multiply: 45⋅5=22545 \cdot 5 = 22545⋅5=225.Now, we have a46=2+225a_{46} = 2 + 225a46​=2+225.

STEP 10

Finally, we add: 2+225=2272 + 225 = 2272+225=227.So, the 46th term is **227**!

STEP 11

The 46th term of the sequence is 227\boxed{227}227​.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionThe first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary). $2,7,12, \ldots$
Find the 46 th term.

What is this asking? We're figuring out what the 46th number is in a list of numbers that starts with 2, 7, and 12. Watch out! Don't just add 5 to 12 forty-three times!
There's a slicker way.

1. Find the Pattern
2. Create the Formula
3. Calculate the 46th Term

Alright, let's look closely at these numbers: 2, 7, and 12.
What's happening here?

From 2 to 7, we're adding 5.
From 7 to 12, we're also adding 5!
So, this looks like an arithmetic sequence with a common difference of $5$ .

The formula for the nth term of an arithmetic sequence is $a_n = a_1 + (n-1) \cdot d$ , where $a_n$ is the term we're looking for, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

In our case, the first term $a_1$ is $2$ , and the common difference $d$ is $5$ .
We want to find the 46th term, so $n = 46$ .
Let's plug these values into our formula!

Substituting our values, we get $a_{46} = 2 + (46-1) \cdot 5$ .

First, inside the parentheses, $46 - 1 = 45$ .
So, our formula becomes $a_{46} = 2 + 45 \cdot 5$ .

Next, we multiply: $45 \cdot 5 = 225$ .
Now, we have $a_{46} = 2 + 225$ .

Finally, we add: $2 + 225 = 227$ .
So, the 46th term is 227!

The 46th term of the sequence is $\boxed{227}$ .