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

Find the sum of an infinite number of terms of a geometric sequence given the first term is 26/5 and the fourth term is βˆ’650/343.

03:31

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‘Ÿ. We see that this tells us our first term is just the constant π‘Ž and our fourth term is this constant π‘Ž multiplied by the ratio π‘Ÿ cubed.

We can use this to find the ratio π‘Ÿ of the geometric sequence given to us in the question. The first term is 26 over five. So we’ll set π‘Ž equal to 26 over five. And our fourth term is equal to negative 650 divided by 343. So this is equal to π‘Ž multiplied by π‘Ÿ cubed. We can substitute π‘Ž 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‘Ÿ cubed.

Now we’ll multiply both sides for our equation by five and divide by 26. This gives us that π‘Ÿ 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‘Ÿ cubed.

Finally, we can take the cube roots of both sides of our equation to get that π‘Ÿ 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‘Ž and ratio of successive terms π‘Ÿ, if the absolute value of π‘Ÿ is less than one. Then the sum from 𝑛 equals zero to ∞ of π‘Ž multiplied by π‘Ÿ to the 𝑛th power is equal to π‘Ž divided by one minus π‘Ÿ.

We’ve already shown that the initial term of our geometric series π‘Ž is equal to 26 over five and the ratio of successive terms π‘Ÿ is equal to negative five over seven. And the absolute value of our ratio π‘Ÿ 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‘Ÿ is less than one, we can calculate the sum of an infinite number of terms of our geometric series by using the formula π‘Ž divided by one minus π‘Ÿ. Using this, we have the sum of an infinite number of terms of our geometric series is equal to the sum from 𝑛 equals zero to ∞ of 26 over five multiplied by negative five over seven all raised to the 𝑛th 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, we’re 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, we’ve 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.

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