Question Video: Finding the Sum of an Infinite Geometric Sequence given the Values of Two Terms

Find the sum of an infinite geometric sequence given the first term is 171 and the fourth term is 171/64.

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Video Transcript

Find the sum of an infinite geometric sequence given the first term is 171 and the fourth term is 171 over 64.

We know that we can find the sum of a convergent geometric sequence by using the formula sum to โˆž is ๐‘Ž over one minus ๐‘Ÿ. And the sequence is said to be convergent if the absolute value of its ratio is less than one. Now, ๐‘Ž is the first term in the sequence, and weโ€™re told that the first term is 171. But what is the common ratio? Well, weโ€™ll use the general formula for the ๐‘›th term in a geometric sequence to find this.

This is ๐‘Ž times ๐‘Ÿ to the power of ๐‘› minus one, where once again ๐‘Ž is the first term and ๐‘Ÿ is the common ratio. Weโ€™ll combine this with the fact that the fourth term in the sequence is 171 over 64. And this means that ๐‘ข sub four is 171, thatโ€™s ๐‘Ž, times ๐‘Ÿ to the power of four minus one or ๐‘Ÿ cubed. But in fact, we know the value of ๐‘ข sub four. Itโ€™s 171 over 64. And so, our equation is 171 over 64 equals 171 times ๐‘Ÿ cubed. Letโ€™s divide both sides of this equation by 171. And so, ๐‘Ÿ cubed is equal to one over 64.

We can solve for ๐‘Ÿ by taking the cube root of both sides, and we find that ๐‘Ÿ is the cube root of one over 64, which is simply equal to one-quarter. The absolute value of one-quarter is just one-quarter; itโ€™s less than one. And so, weโ€™ve confirmed that the geometric sequence is indeed convergent. And we now know the value of ๐‘Ž, thatโ€™s 171, and ๐‘Ÿ.

Letโ€™s substitute everything we know into the formula. We get 171 over one minus one-quarter as being the sum to โˆž, in other words, the sum of all terms in our sequence. Thatโ€™s 171 over three-quarters. Now, of course, we can divide by a fraction by multiplying by the reciprocal of that fraction. Thatโ€™s the same as 171 times four over three. Then, we cross-cancel by dividing 171 and three by three, meaning weโ€™re left with 57 times four over one, which is just 57 times four. And thatโ€™s equal to 228. And we can, therefore, say that the sum of an infinite geometric sequence with a first term of 171 and a fourth term of 171 over 64 is 228.

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