In Exercises 17–24, write a formula for the general term (the nth...

Question

In Exercises 17–24, write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a7, the seventh term of the sequence.

3, 12, 48, 192, ...

Accepted Answer

Hello. Today we're going to be finding a formula that describes the term of the geometric sequence. And we're going to use that formula to find the eighth term of the sequence. Now, in order to find the formula for the nth term. We're going to use the general formula for the nth term and that is given to us as a sub N is equal to our first term, A sub one times our common ratio are raised to the power of n minus one. Now, in order to fill this out, we're going to need two things. We're going to need our first term and we're going to need our common ratio. Well, the first term in our given geometric sequence is going to be 64. So we're going to allow a sub 12 equal to 64. Next we need to find the common ratio and the common ratio is any term in the geometric sequence divided by its previous term. So here we can take the second term a sub two and divided by the first term, a sub one. The second term in the sequence is 96. So taking this division is going to give us 96 over a sub one which is 64 that's going to simplify to 3/2. Now let's go ahead and plug this into the general formula for the nth term. Doing this is going to give us a sub N is equal to our first term which is 64 times the common ratio which is three over to raise to the power of n minus one. And this is going to be the formula for the nth term of any given geometric sequence or specifically this one. Now we need to use this one to find the eighth term of the geometric sequence. Since we want to find the eighth term, we're going to allow N two equal to eight. And the reason why we let N equal to eight is because N represents the amount of terms that are given in your sequence. So using N equal to eight is going to give us a sub eight is equal to 64 times three over to raise to the power of n minus one which is eight minus one and eight minus one is going to give us seven. So we have 64 times three over to race the power of seven. Now using our exponential rule, we're going to distribute this exponent of seven to both the numerator and denominator and that's going to give us 64 times three to the power of 7/2 to the power of seven Evaluating this is going to give us 64 times 2187 over Multiplying the values together is going to give us 139, 968 over and this is going to give us the fraction over two which has an approximate decimal value of 1093 .5. And this is going to be the eighth term in the geometric sequence. So just to recap our formula for the nth term is a sub N is equal to 64 times three over to raise to the power of n minus one. And we use this to calculate the eighth term, which is 1093.5. With that being said, The answer to this problem is going to be be so I hope this video helps you in understanding how to approach this geometric sequence problem and I'll go ahead and see you all in the next video.

In Exercises 17–24, write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a7, the seventh term of the sequence. 3, 12, 48, 192, ...

Key Concepts

Geometric Sequence

General Term Formula

Finding Specific Terms

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