Question

is 729 correct?

Answer 1

Where is the question?

Answer 2

From what I see the sequence is going by 3's

Answer 3

yeh..

Answer 4

@Vocaloid

Answer 5

3 X 3 = 9
9 X 3 = 27

Answer 6

So, you're correct



\[3\times 1=3 3*2=9 3*3=27 3*4=81 3*5=243 3*6=729\]

Answer 7

sorry, I didn't put spaces in between 3*1

Answer 8

but it goes 3*1=3 3*2 = ect

Answer 9

This is what is known as a geometric sequence. This is where a sequence of numbers is found by multiplying the previous term by a common ratio, known as variable r.

\[a _{2} = a _{_{1}} \times r\]

This is in contrast to an arithmetic sequence. This is where a sequence is found by adding by a common difference, denoted as d.

\[a _{2} = a _{1} + d\]

Where,
\[a _{1} \] is the first term in a sequence, and
\[a _{2}\] is the second term in a sequence.

To solve for the nth term of a geometric sequence, we can use the following formula.

\[a _{n} = a _{1} \times r ^{n - 1}\]

This formula is essentially saying: the nth term of a geometric sequence is equal to the first term of the sequence, multiplied by the common ratio taken to the n-1.

If we look at our given information, we see that our first term (a1) is 1.

Any common ratio for a geometric sequence can be found from the formula:

\[r = \frac{ a _{2}}{ a _{1} }\]

Input:

\[r = \frac{ 3 }{ 1 } = 3\]

This can also work at any point in the sequence

\[r = \frac{ 27 }{ 9 } = 3\]

Now that we know our first term, and our common ratio, we can use our formula.

\[a _{n} = a _{1} \times r ^{n - 1}\]

\[a _{7} = 1 \times 3^{7 - 1}\]
\[a _{7} = 1 \times 3^{6}\]
\[a _{7} = 729\]

Answer 10

You are correct. But moving forward, it is better to use the method with formulas as you will not always be able to multiply it out. For example, what if I asked you to find the 1050th term of a sequence. These concepts help set you up for calculus as well.