In Exercises 1–8, write the first five terms of each geometric sequence.

a1 = 20, r = 1/2

Question

In Exercises 1–8, write the first five terms of each geometric sequence.

a1 = 20, r = 1/2

Accepted Answer

Hello. Today we're going to be finding the first eight terms of a geometric sequence using the given information. So we are told that the first term of the geometric sequence is and we are given the common ratio which is one third. Now, since we don't have a recurrence formula to work with, we need to use the general form of the term of a general of a geometric sequence. And that is given to us as a sub, N is equal to our first term. A sub one times the common ratio are raised to the power of N -1. So all we need is N a sub one and R and R N is going to change depending on what term we're looking for. So since we already given the first term, let's go ahead and start by finding the second term. And since we want to find the second term we're going to allow N two equal to two. So plugging in And is equal to into our general and term formula is going to give us a sub two is equal to the first term which is 27 times the common ratio which is 1/3. Race to the power of N -1 which is 2 -1. Now, two minus one is just going to give us one. So we're not gonna need the exponent here and 27 times one third is going to give us nine. So that's going to be the second term of the geometric sequence. And we're going to repeat this process for the remaining six terms. So next we're gonna let N is equal to three and that's going to give us a sub three is equal to 27 times one. Third race of the power of three minus one which is going to be two. And using our exponential rule, we're going to distribute this exponent of two to both the numerator and denominator. That's going to give us 27 times one to the power of two, which is just 1/3 squared, which is nine and 27 times one. Ninth is going to give us three. Next we're going to let any sequel to four and when N is equal to four, we get a sub four is equal to 27 times one. Third race of the power of four minus one which is going to be three. And once again we're going to distribute this exponent to both the numerator and denominator, that's going to give us 27 times 1/27. And that's going to give us one. Now we're going to find the fifth term when N is equal to five and when N is equal to five we have a sub five is equal to 27 times 1/3 raised to the power of 5 -1 which is four. Again we're distributing this exponent And that's going to give us Times 1/81 And 27 times 1/81 is going to give us 27. Order over 81 which simplifies to 1/3. Next we're gonna let N is equal to six. And when n is equal to six we get a sub six is equal to 27 times 1/3 raised to the power of six minus one which is five, distribute the exponents. And we get 27 times 1/3 to the power of five which is going to be And 27 times 1/2 43 is going to give us 27 over 243 which simplifies to 1 9th. Next we're gonna let it is equal to seven and that gives us a sup seven is equal to 27 times 1/3 races. The power of seven minus one which is six, Distributing the exponents, is going to give us 27 Times 1/3 to the power of six which is 729. And multiplying 27 with one over 729 is going to give us 27 over 729 which reduces to 1/27. And finally we're going to let N is equal to eight. And that's going to give us a sub eight is equal to 27 times one. Third race of the power of eight minus one which is seven. We're going to distribute the exponent. And that's going to give us 27 Times 1/3 to the power of seven, which gives us 2187 multiplying this together is going to give us 27 over 2187 which simplifies to 1/81. And that's going to be the final term in the geometric sequence. Now, let's go ahead and just write out all of our terms. We were told that the first term is 27 then we calculated the rest. The next term was nine, then three, one, one, third, 1/9 1/27 and 1/81. And this is going to be the first eight terms of the geometric sequence with the given common ratio and the given first term. And with that being said, the answer to this problem is going to be D. 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 1–8, write the first five terms of each geometric sequence. a1 = 20, r = 1/2

Verified Solution

Key Concepts

Geometric Sequence

First Term and Common Ratio

Finding Terms of a Sequence