Question

I need some help solving this question please! Thank you for the help!

Answer 1

ANSWERS

• Common ratio: 4

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• Formulas

`\begin{gathered} explicit._{}_{}formula\colon a_n=3(4)^(n-1) \\ revursive.formulaa\colon a_n=4a_(n-1) \end{gathered}`

• Next 3 terms: 768, 3072, 12288

• 8th term: 49152

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• 9th term: 196608

Step-by-step explanation

The common ratio is the number by which we have to multiply a term of the sequence to obtain the next. In this sequence, we can find the common ratio by dividing each term by the previous term,

`\begin{gathered} 12\colon3=4 \\ 48\colon12=4 \\ 192\colon48=4 \end{gathered}`

Hence, the common ratio is 4.

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

`a_n=a_1\cdot r^(n-1)`

Where r is the common ratio and a1 is the first term. For this sequence, a1 = 3 and r = 4, so the explicit formula is,

`a_n=3(4)^(n-1)`

The recursive formula for the nth term of a geometric sequence is,

`a_n=r\cdot a_(n-1)`

So for this sequence,

`a_n=4a_(n-1)`

To find the next 3 terms it will be easier if we use the recursive formula so that we don't have to think about what number of terms is the last one we have. So if the last term we have is 192, the next one is,

`a_{\text{next}1}=4\cdot192=768`

And the next two terms are,

`\begin{gathered} a_{\text{next}2}=4\cdot768=3072 \\ a_{\text{next}3}=4\cdot3072=12288 \end{gathered}`

Hence, the next 3 terms are 768, 3072, 12288.

To find the 8th and 9th terms we use the explicit formula because we just have to replace n by 8 or 9 respectively and solve,

`a_8=3\cdot4^(8-1)=3\cdot4^7=3\cdot16384=49152`
`a_9=3\cdot4^(9-1)=3\cdot4^8=3\cdot65536=196608`

Hence, the 8th term is 49152 and the 9th term is 196608.