### Video Transcript

Find the sum of an infinite number
of terms of a geometric sequence given the first term is 26 over five and the fourth
term is negative 650 divided by 343.

The question gives us a geometric
sequence with first term 26 over five and fourth term negative 650 divided by
343. We recall that geometric sequence
has the property that, to get the next term in the sequence, we multiply the
previous term by a constant ratio π. We see that this tells us our first
term is just the constant π and our fourth term is this constant π multiplied by
the ratio π cubed.

We can use this to find the ratio
π of the geometric sequence given to us in the question. The first term is 26 over five. So weβll set π equal to 26 over
five. And our fourth term is equal to
negative 650 divided by 343. So this is equal to π multiplied
by π cubed. We can substitute π is equal to 26
over five into our equation for the fourth term. This gives us that negative 650
divided by 343 is equal to 26 over five multiplied by π cubed.

Now weβll multiply both sides for
our equation by five and divide by 26. This gives us that π cubed is
equal to negative 650 multiplied by five divided by 343 multiplied by 26. This simplifies to give us negative
125 divided by 343 is equal to π cubed.

Finally, we can take the cube roots
of both sides of our equation to get that π is equal to the cube root of negative
125 divided by 343, which is equal to negative five over seven.

Now the question wants us to find
the sum of an infinite number of terms of our geometric sequence. We recall that, for a geometric
series with initial value π and ratio of successive terms π, if the absolute value
of π is less than one. Then the sum from π equals zero to
β of π multiplied by π to the πth power is equal to π divided by one minus
π.

Weβve already shown that the
initial term of our geometric series π is equal to 26 over five and the ratio of
successive terms π is equal to negative five over seven. And the absolute value of our ratio
π is the absolute value of negative five over seven, which is just equal to five
over seven, which is less than one.

So since the absolute value of our
ratio π is less than one, we can calculate the sum of an infinite number of terms
of our geometric series by using the formula π divided by one minus π. Using this, we have the sum of an
infinite number of terms of our geometric series is equal to the sum from π equals
zero to β of 26 over five multiplied by negative five over seven all raised to the
πth power. And this is equal to 26 over five
divided by one minus negative five over seven.

We can rewrite our denominator as
seven over seven plus five over seven, which is just equal to 12 divided by
seven. Next, instead of dividing by the
fraction 12 over seven, weβre going to multiply by the reciprocal. This gives us 26 over five
multiplied by seven divided by 12. We can cancel out a shared factor
of two in the numerator and the denominator. Finally, we can evaluate this to be
equal to 91 divided by 30.

Therefore, weβve shown that the sum
of an infinite number of terms of the geometric series with first term 26 over five
and fourth term negative 650 divided by 343 is equal to 91 divided by 30.