### 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 โ is ๐ over one minus
๐. And the sequence is said to be
convergent if the absolute value of its ratio is less than one. Now, ๐ is the first term in the
sequence, and weโre told that the first term is 171. But what is the common ratio? Well, weโll use the general formula
for the ๐th term in a geometric sequence to find this.

This is ๐ times ๐ to the power of
๐ minus one, where once again ๐ is the first term and ๐ is the common ratio. Weโll combine this with the fact
that the fourth term in the sequence is 171 over 64. And this means that ๐ข sub four is
171, thatโs ๐, times ๐ to the power of four minus one or ๐ cubed. But in fact, we know the value of
๐ข sub four. Itโs 171 over 64. And so, our equation is 171 over 64
equals 171 times ๐ cubed. Letโs divide both sides of this
equation by 171. And so, ๐ cubed is equal to one
over 64.

We can solve for ๐ by taking the
cube root of both sides, and we find that ๐ 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; itโs less than one. And so, weโve confirmed that the
geometric sequence is indeed convergent. And we now know the value of ๐,
thatโs 171, and ๐.

Letโs substitute everything we know
into the formula. We get 171 over one minus
one-quarter as being the sum to โ, in other words, the sum of all terms in our
sequence. Thatโs 171 over three-quarters. Now, of course, we can divide by a
fraction by multiplying by the reciprocal of that fraction. Thatโs the same as 171 times four
over three. Then, we cross-cancel by dividing
171 and three by three, meaning weโre left with 57 times four over one, which is
just 57 times four. And thatโs 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.