### Video Transcript

Find the infinite geometric
sequence given the first term is five over 57 and the sum of the terms is 20 over
171.

We have an infinite geometric
sequence with first term five over 57 and sum of the terms 20 over 171. We know that the general form of a
geometric sequence is π one, π one π, π one π squared, and so on, where π one
is the first term of the sequence, which in this case is five over 57, and π is a
common ratio. Itβs what we multiply by to get the
next term.

Another really useful formula that
we have is the sum of a number of terms of an infinite geometric sequence, which is
given by π equals π one over one minus π. And this is only valid when the
absolute value of π is less than one. In order to find what this
geometric sequence is starting with the first term five over 57, weβre going to need
to know what we need to multiply by to get each next term. And thatβs where this formula is
going to come in handy.

We know what π is. Itβs the sum of all the terms in
the infinite geometric sequence. And weβre given in the question
that this is 20 over 171. We also know π one. Itβs the first term of the
sequence, and weβre told this is five over 57. But we donβt know what π is. So letβs rearrange this for π.

We can start by multiplying both
sides of the equation by one minus π. And then we can divide both sides
by 20 over 171. So we have that one minus π equals
five over 57 over 20 over 171. We could think of this as five over
57 divided by 20 over 171. And from our rules of dividing
fractions, we know that this is the same as five over 57 times 171 over 20, which,
by multiplying the numerators and multiplying the denominators, we find to be 855
over 1140, which cancels down to three over four.

So we have that one minus π is
equal to three over four, which leads us to π equals one over four. And notice how the absolute value
of π is less than one, as we said it needs to be to use the sum formula. Remember, we said that π is the
thing that we multiply by to find the next term in the sequence. So starting with the first term as
five over 57, we find the next term by multiplying five over 57 by one over
four. This gives us that the second term
is five over 228. Then to find the third term, we
multiply the second term by one over four. So we do five over 228 multiplied
by one over four. And this gives us five over
912. And then, the sequence continues in
the same way, by multiplying each term by one over four to find the next term.

So by using the formula for the sum
of an infinite geometric sequence, we were able to determine what the common ratio
is and therefore find the first few terms of this infinite geometric sequence.