QuestionFind the $n$ th term of the sequence: $198 + (n+4) \cdot 2^{31}$ .

Studdy Solution

STEP 1

Assumptions1. The sequence is arithmetic, meaning that each term is obtained by adding a constant difference to the previous term. . The first term of the sequence is $198+5 \cdot^{31}$ .
3. The second term of the sequence is $198+6 \cdot^{31}$ .
4. The third term of the sequence is $198+7 \cdot^{31}$ .
5. We are asked to find the $n$ th term of this sequence.

STEP 2

The common difference of the sequence can be obtained by subtracting the first term from the second term or the second term from the third term. Let's calculate the common difference.
$d = (198+6 \cdot2^{31}) - (198+5 \cdot2^{31})$

STEP 3

implify the expression to find the common difference.
$d =2^{31}$

STEP 4

The $n$ th term of an arithmetic sequence can be found using the formula $a_n = a1 + (n-1) \cdot d$ where $a_n$ is the $n$ th term, $a1$ is the first term, $d$ is the common difference, and $n$ is the term number.

STEP 5

Substitute the values of $a1$ , $d$ , and $n$ into the formula.
$a_n = (198+5 \cdot2^{31}) + (n-1) \cdot2^{31}$

STEP 6

implify the expression to find the $n$ th term.
$a_n =198 + (5+n-1) \cdot2^{31}$

STEP 7

Further simplify the expression.
$a_n =198 + (4+n) \cdot2^{31}$ The $n$ th term of the sequence is $198 + (4+n) \cdot2^{31}$ .

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