QuestionFind the formula for the $n^{\text{th}}$ term $a_{n}$ of the sequence: 6, 16, 26, ...

Studdy Solution

STEP 1

Assumptions1. The sequence is arithmetic, meaning the difference between consecutive terms is constant. . The first term of the sequence is6.
3. The common difference, d, is the difference between the second term and the first term.

STEP 2

First, we need to find the common difference, d. We can do this by subtracting the first term from the second term.
$d = a2 - a1$

STEP 3

Now, plug in the given values for the first and second terms to calculate the common difference.
$d =16 -6$

STEP 4

Calculate the common difference.
$d =16 -6 =10$

STEP 5

Now that we have the common difference, we can write the general formula for an arithmetic sequence.
$a_n = a1 + (n -1) \cdot d$

STEP 6

Plug in the values for the first term and the common difference into the formula.
$a_n =6 + (n -1) \cdot10$

STEP 7

implify the formula.
$a_n =6 +10n -10$

STEP 8

Combine like terms.
$a_n =10n -4$ The explicit formula for the nth term of the sequence is $a_n =10n -4$ .

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QuestionFind the formula for the $n^{\text{th}}$ term $a_{n}$ of the sequence: 6, 16, 26, ...

Studdy Solution

STEP 1

Assumptions1. The sequence is arithmetic, meaning the difference between consecutive terms is constant. . The first term of the sequence is6.
3. The common difference, d, is the difference between the second term and the first term.

STEP 2

First, we need to find the common difference, d. We can do this by subtracting the first term from the second term.
$d = a2 - a1$

STEP 3

Now, plug in the given values for the first and second terms to calculate the common difference.
$d =16 -6$

STEP 4

Calculate the common difference.
$d =16 -6 =10$

STEP 5

Now that we have the common difference, we can write the general formula for an arithmetic sequence.
$a_n = a1 + (n -1) \cdot d$

STEP 6

Plug in the values for the first term and the common difference into the formula.
$a_n =6 + (n -1) \cdot10$

STEP 7

implify the formula.
$a_n =6 +10n -10$

STEP 8

Combine like terms.
$a_n =10n -4$ The explicit formula for the nth term of the sequence is $a_n =10n -4$ .

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QuestionFind the formula for the nthn^{\text{th}}nth term ana_{n}an​ of the sequence: 6, 16, 26, ...

Studdy Solution

STEP 1

Assumptions1. The sequence is arithmetic, meaning the difference between consecutive terms is constant. . The first term of the sequence is6.3. The common difference, d, is the difference between the second term and the first term.

STEP 2

First, we need to find the common difference, d. We can do this by subtracting the first term from the second term.d=a2−a1d = a2 - a1d=a2−a1

STEP 3

Now, plug in the given values for the first and second terms to calculate the common difference.d=16−6d =16 -6d=16−6

STEP 4

Calculate the common difference.d=16−6=10d =16 -6 =10d=16−6=10

STEP 5

Now that we have the common difference, we can write the general formula for an arithmetic sequence.an=a1+(n−1)⋅da_n = a1 + (n -1) \cdot dan​=a1+(n−1)⋅d

STEP 6

Plug in the values for the first term and the common difference into the formula.an=6+(n−1)⋅10a_n =6 + (n -1) \cdot10an​=6+(n−1)⋅10

STEP 7

implify the formula.an=6+10n−10a_n =6 +10n -10an​=6+10n−10

STEP 8

Combine like terms.an=10n−4a_n =10n -4an​=10n−4The explicit formula for the nth term of the sequence is an=10n−4a_n =10n -4an​=10n−4.

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QuestionFind the formula for the $n^{\text{th}}$ term $a_{n}$ of the sequence: 6, 16, 26, ...

Assumptions1. The sequence is arithmetic, meaning the difference between consecutive terms is constant. . The first term of the sequence is6.
3. The common difference, d, is the difference between the second term and the first term.

First, we need to find the common difference, d. We can do this by subtracting the first term from the second term.
$d = a2 - a1$

Now, plug in the given values for the first and second terms to calculate the common difference.
$d =16 -6$

Calculate the common difference.
$d =16 -6 =10$

Now that we have the common difference, we can write the general formula for an arithmetic sequence.
$a_n = a1 + (n -1) \cdot d$

Plug in the values for the first term and the common difference into the formula.
$a_n =6 + (n -1) \cdot10$

implify the formula.
$a_n =6 +10n -10$

Combine like terms.
$a_n =10n -4$ The explicit formula for the nth term of the sequence is $a_n =10n -4$ .