Write a formula for the general or nthn^{\th}nth term of the geometric sequence where a7=1458a_7=1458a7=1458 and r=−3r=-3r=−3.

a n = 2 ⋅ ( − 3 ) n − 1 a_{n}=2\cdot\left(-3\right)^{n-1} a n = 2 ⋅ ( − 3 ) n − 1

Write a formula for the general or n th n^{\th} n t h term of the geometric sequence where a 7 = 1458 a_7=1458 a 7 = 1458 and r = − 3 r=-3 r = − 3 .

Everyone. So in this practice problem, we're gonna find a general formula or a formula for the general or nth term of a geometric sequence. And the information that we're given here says that a seven is equal to a 458, and r is equal to negative three. Let's go ahead and get started here. Remember the that the general form for a geometric sequence is a n equals a one times r to the n minus one power. So if I wanna figure out a general formula for this, I'm just gonna write that out. A n equals a one times r to the n minus one. What I need from this equation in order in order to write it is I need to know what the first term is, and I need to know what the common ratio is. So let's do the inform let's look through the information that we're given, and we'll see that we actually have one of those variables already. We have r equals negative three. So I have r. So can I just start writing this equation? Well, not quite, because what we're told is that a seven equals, fourteen fifty eight. We actually have no idea what the first term is. What is a one? So in order to figure this out, what I'm gonna have to do over here is I'm gonna have to figure out what the first term of the sequence is. How do I do that? Well, let's just look at the information that I do know. I know that a seven is equal to a 450 eights. So what I can use here is I can use the fact that a seven, and I'm gonna just sort of use the general formula, is just equal to a one times r, in which case, this is just gonna be negative three, to the n minus one power, which is gonna be six in this case. Right? So here, I'm just looking. I'm just using the fact that n is equal to seven. So this number right here, when I plug this in, even though I don't know what a one is equal to, I know that the seventh term is equal to a thousand 458. So what I can use here is I can just use these, these numbers to actually figure out what a one is equal to. So go ahead and punch negative three to the sixth power into your calculators, and you'll see that this, simplifies down to a one times and this is gonna be 729. Make sure when you plug it into your calculator that you're doing negative three in parentheses to the sixth power, because if you do it without the parentheses like this, you're gonna get a negative number. So be really careful when you do that. This is 729, and this is gonna be a 458. Now it's just pretty simple division. Right? So we're gonna divide 729 on each side, cancels out, and what you're gonna find here is that a one is equal to two. So now that we know what the first term is, we can plug it back into our general formula. Alright? So we look for general formulas just to recap. We actually used what one of the terms was. We know what the number is. We used the general formula to figure out what the first term is by using that equation, and now we can plug it in and find what the general formula is. Alright. So I'm just gonna plug this right back into this formula over here. And I have that a n is equal to two times negative three to the n minus one power. This is the general formula, and with this equation, I could figure out any term in the sequence, any nth term. Alright, folks. So that's it for this one. Let me know if you have any questions. Thanks for watching.

Write a formula for the general or n th ⁡ n^{\th} n t h term of the geometric sequence where a 7 = 1458 a_7=1458 a 7 = 1458 and r = − 3 r=-3 r = − 3 .

a n = 2 ⋅ ( − 3 ) n − 1 a_{n}=2\cdot\left(-3\right)^{n-1} a n = 2 ⋅ ( − 3 ) n − 1

a n = 1 ⋅ ( − 3 ) n − 1 a_{n}=1\cdot\left(-3\right)^{n-1} a n = 1 ⋅ ( − 3 ) n − 1

a n = − 2 3 ⋅ ( − 3 ) n − 1 a_{n}=-\frac23\cdot\left(-3\right)^{n-1} a n = − 3 2 ⋅ ( − 3 ) n − 1

a n = 2 3 ⋅ ( − 3 ) n − 1 a_{n}=\frac23\cdot\left(-3\right)^{n-1} a n = 3 2 ⋅ ( − 3 ) n − 1

14. Sequences & Series

Sequences

Business Calculus

14. Sequences & Series

Sequences

Struggling with Business Calculus?

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Multiple Choice

Write a formula for the general or $n^{\th}$ term of the geometric sequence where $a_7=1458$ and $r=-3$ .

$a_{n}=1\cdot\left(-3\right)^{n-1}$

$a_{n}=2\cdot\left(-3\right)^{n-1}$

$a_{n}=-\frac23\cdot\left(-3\right)^{n-1}$

$a_{n}=\frac23\cdot\left(-3\right)^{n-1}$

Verified step by step guidance

Step 1: Recall the general formula for the nth term of a geometric sequence:

a_{n} = a r_{n} = a r_{n}

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