QuestionFind the nth term of the sequence $7, 7^{4}, 7^{7}, 7^{10}, \ldots$ and simplify your answer.

Studdy Solution

STEP 1

Assumptions1. The sequence is $7,7^{4},7^{7},7^{10}, \ldots$ . The sequence is in the form of $7^{n}$ where $n$ is a term in the sequence $1,4,7,10, \ldots$
3. The nth term of the sequence is to be found

STEP 2

First, we need to find the nth term of the sequence $1,4,7,10, \ldots$ . This is an arithmetic sequence with a common difference of.
The nth term of an arithmetic sequence is given by the formula $a_n = a1 + (n-1)d$ where $a_n$ is the nth term, $a1$ is the first term, $n$ is the term number, and $d$ is the common difference.

STEP 3

Plug in the values for the first term and the common difference to find the nth term of the sequence $1,,7,10, \ldots$ .
$a_n =1 + (n-1)3$

STEP 4

implify the expression to get the nth term.
$a_n =1 +3n -3$ $a_n =3n -2$

STEP 5

Now that we have the nth term of the sequence $1,4,7,10, \ldots$ , we can find the nth term of the sequence $7,7^{4},7^{7},7^{10}, \ldots$ .
The nth term of this sequence is $7^{a_n}$ where $a_n$ is the nth term of the sequence $1,4,7,10, \ldots$ .

STEP 6

Plug in the value for $a_n$ to find the nth term of the sequence $,^{4},^{},^{10}, \ldots$ .
$^{a_n} =^{3n -2}$ The nth term of the sequence $,^{4},^{},^{10}, \ldots$ is $^{3n -2}$ .

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