Question

6) Given the geometric sequence 2, 4, 8, 16,...

a) Determine the common ratio.

b) Write the explicit formula.

c) Find a10

d) Write the recursive formula.

Answer 1

a) Common ratio r =
`\boxed{2}`

b) Formula for common ratio:

`(a_(n+1))/(a_n) , n = 1, 2, 3, 4....\\`

Formula for nth term:

`a_n = a_1r^(n-1)`

(see explanation below for part b)

c)
`a_(10) = \boxed{1024}`

d) Recursive formula:
`\boxed{a_(n+1) = 2a_n}`

Explanation:

a) The common ratio r for a geometric sequence is the ratio of any term(except the first) to the previous term. And this is a constant for that sequence

If we have a sequence
`a_1, a_2, a_3, a_4, a_5,....\\`

`r = (a_2)/(a_1) = (a_3)/(a_2) = (a_4)/(a_3) = (a_5)/(a_4)`

Here
`a_1` is the first term in the sequence

The sequence we have here is
2, 4, 8, 16

So

`r = (4)/(2) = (8)/(4) = (16)/(8) = 2\\`
Answer to a)

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b) This part is confusing. Are they asking for a formula for the common ratio or a formula for the nth term? You can write both or check with your teacher

The explicit formula for the common ratio is

`(a_(n+1))/(a_n) , n = 1, 2, 3, 4....\\`

The explicit formula for the nth term,
`a_n` of a geometric sequence is

`a_n = a_1r^(n-1)`

where
`a_1` is the first term of the sequence and r is the common ratio
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c)
The formula for finding the nth term of a geometric sequence is

`a_n = a_1r^(n-1)\\\\`

Therefore

`a_(10) = a_1 r^((10-1)) = a_1 * r^9\\\\`

`a_1 = 2, r = 2\\\\`

So

`a_(10) = 2 \cdot 2^9 \\\\= 2 \cdot 512\\\\= 1024\\`ANSWER c)

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d)

A recursive formula is just an expression that denotes the relationship between one term and the next

We know that

`r= (a_(n+1))/(a_n) \\\\`

Multiplying both sides by
`a_n`,

`ra_n = a_(n+1)`

Or

`a_(n+1) = ra_n\\\\`

For the specific case of r = 2 we get the recursive formula as

`a_(n+1) = 2a_n`

ANSWER d)