Question Video: Finding a Geometric Sequence and the Sum of 𝑛 Terms given the Sum of Another 𝑛 Terms | Nagwa Question Video: Finding a Geometric Sequence and the Sum of 𝑛 Terms given the Sum of Another 𝑛 Terms | Nagwa

Question Video: Finding a Geometric Sequence and the Sum of 𝑛 Terms given the Sum of Another 𝑛 Terms Mathematics • Second Year of Secondary School

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Find the geometric sequence and the sum of the first five terms given the sum of the first three terms is 1 and the sum of the next three terms is 27.

06:12

Video Transcript

Find the geometric sequence and the sum of the first five terms given the sum of the first three terms is one and the sum of the next three terms is 27.

In this question, we’re asked to find two things. First, we’re asked to find a geometric sequence, and we’re also asked to find the sum of the first five terms of this geometric sequence. To do this, we’re given two pieces of information. We’re told the sum of the first three terms of this geometric sequence is one, and we’re also told the sum of the next three terms of this geometric sequence is 27.

To answer this question, we start by recalling a geometric sequence is a sequence where the ratio between successive terms is constant. And this means to generate terms in our sequence, we multiply by a constant ratio π‘Ÿ. For example, if the first term of our sequence is π‘Ž, the second term will be π‘Ž times π‘Ÿ and the third term will be π‘Žπ‘Ÿ multiplied by π‘Ÿ, which is π‘Žπ‘Ÿ squared. And this pattern continues. The first six terms of a geometric sequence with initial term π‘Ž and ratio of successive terms π‘Ÿ is given as follows.

Therefore, to answer the first part of this question of determining the geometric sequence, we need to determine the value of π‘Ž, the first term of this sequence, and the ratio of successive terms π‘Ÿ. To do this, we need to use the two pieces of information we’re given. First, we’re told the sum of the first three terms is one. And there’s a few different ways of doing this. We could just add the first three terms of this geometric sequence together. However, we’ll do this by using a fact about summing finite geometric sequences.

The sum of the first 𝑛 terms of a geometric sequence with first term π‘Ž and ratio of successive terms π‘Ÿ, 𝑆 sub 𝑛, is equal to π‘Ž multiplied by one minus π‘Ÿ to the 𝑛th power all divided by one minus π‘Ÿ provided π‘Ÿ is not equal to one. And to apply this in our question, we do need to check if the ratio is equal to one. However, we can see that this is not true. If the ratio π‘Ÿ is just equal to one, every single term in the geometric sequence is equal to π‘Ž. And in this case, the first three terms of the geometric sequence would all be π‘Ž and the next three terms of the geometric sequence would also be π‘Ž. And therefore, the sum of the first three terms would be equal to the sum of the next three terms. This is not the case in this question, so π‘Ÿ is not equal to one.

Therefore, we can apply this formula to the sequence. Since the sum of the first three terms of our sequence is equal to one, one must be equal to π‘Ž multiplied by one minus π‘Ÿ cubed all divided by one minus π‘Ÿ. We’re also told the sum of the next three terms of this geometric sequence is 27. And there’s a few different ways of using this. We could just add the three terms of this sequence together and set this equal to 27. We could also use the fact that the sum of these three terms will be the sum of the first six terms of this sequence, where we remove the sum of the first three terms of this sequence. And both of these methods would work, and we can use them if we prefer. However, we’re going to use a different method.

We can see that these three terms in this geometric sequence form their own geometric sequence. The ratio of successive terms is π‘Ÿ and the first term is π‘Žπ‘Ÿ cubed. And π‘Ÿ is not equal to one, so we can add them together using our formula. The first term is π‘Žπ‘Ÿ cubed. And we’re adding three terms of a geometric sequence together, and we know this is equal to 27. This gives us 27 is equal to π‘Žπ‘Ÿ cubed multiplied by one minus π‘Ÿ cubed all divided by one minus π‘Ÿ. We now have two equations in two unknowns. We need to solve this to find the values of π‘Ž and π‘Ÿ.

And there’s many different ways of solving these equations. We’ll only go through one of these. We’re going to multiply the top equation through by 27. On the left-hand side, we’ll get 27, and on the right-hand side, we’ll get an extra factor of 27. This gives us the following equation. Now, the left-hand side of both of our equations are equal. So the right-hand sides of these two equations must also be equal. And if we equate the right-hand side of both of these two equations, we get the following equation. And we can now solve this for the value of π‘Ÿ.

First, remember when we considered our sequence, we were able to show that the value of π‘Ÿ was not equal to one. Therefore, we can multiply both sides of this equation through by one minus π‘Ÿ. We can also divide both sides of the equation through by one minus π‘Ÿ cubed. Next, we want to divide both sides of our equation through by π‘Ž. However, to do this, we need to check if π‘Ž is equal to zero. And in fact we can show that π‘Ž is nonzero in exactly the same way we did to show that π‘Ÿ was not equal to one. If π‘Ž was zero, every single term in our geometric sequence would be equal to zero. And in this case, the sum of the first three terms will be zero. Therefore, π‘Ž cannot be equal to zero. And if π‘Ž is not equal to zero, we can divide both sides of our equation through by π‘Ž, giving us that 27 is equal to π‘Ÿ cubed.

Finally, we take the cube root of both sides of this equation to see that π‘Ÿ is equal to three. We still need to find the value of π‘Ž. We can do this by substituting π‘Ÿ is equal to three into either of our original equations. For example, we can substitute π‘Ÿ is equal to three into the top equation. We get 27 is equal to 27π‘Ž multiplied by one minus three cubed all divided by one minus three. And if we evaluate and rearrange this equation for π‘Ž, we would find that π‘Ž is equal to one divided by 13.

Now that we’ve determined the initial term of the sequence and the ratio of successive terms, we have enough to determine all of the terms of the geometric sequence. The first term π‘Ž is one over 13 and the ratio of successive terms π‘Ÿ is equal to three. So we multiply one over 13 by factors of three to generate the terms of the sequence. π‘Ž sub 𝑛 is one over 13, three over 13, nine over 13, and the sequence continues.

The second part of this question wants us to determine the sum of the first five terms of this sequence. And there’s a few different ways we could do this. We could just find the first five terms of the sequence and add them together, or we could just use our formula. Substituting π‘Ž is equal to one over 13, π‘Ÿ is equal to three, and 𝑛 is equal to five into this formula, we get 𝑆 sub five is equal to one over 13 multiplied by one minus three to the fifth power all divided by one minus three, which if we evaluate we’ll see is equal to 121 divided by 13, which gives us our final answer.

If we have a geometric sequence where the sum of the first three terms is one and the sum of the next three terms is 27, then we were able to show the geometric sequence π‘Ž sub 𝑛 would be one over 13, three over 13, nine over 13, and the sequence continues. And the sum of the first five terms 𝑆 sub five would be 121 divided by 13.

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