Find the infinite geometric sequence given each term of the sequence is twice the sum of the terms that follow it and the first term is 37.
The question asks us to find an infinite geometric sequence. We recall that a sequence of initial term 𝑎 and ratio of successive terms 𝑟 — so that’s 𝑎, then 𝑎𝑟, then 𝑎𝑟 squared, and this goes on infinitely — is an infinite geometric sequence. So the question is asking us to find a sequence of this form. And it gives us some properties of this sequence. It tells us that the first term of this sequence is 37. So the value of 𝑎 in our infinite geometric sequence is 37.
The last piece of information the question tells us is that every term of our sequence is twice the sum of the terms that follow it. Well, what does this mean? Let’s call the 𝑛th term of our infinite geometric sequence 𝑎𝑛. So this is equal to 𝑎 multiplied by 𝑟 to the 𝑛th power for some values 𝑎 and 𝑟. We already know that 𝑎 is equal to 37. We’re told that every term is two times the sum of all the successive terms. So in particular, the first term is equal to two multiplied by the sum of all the rest of the terms.
Our first term is 37, and this is equal to two multiplied by 𝑎 one plus 𝑎 two plus 𝑎 three. And this goes on and on. And this is equal to 37𝑟 plus 37𝑟 squared plus 37𝑟 cubed. And this keeps going.
Now, we could simplify the right-hand side of our equation by using the infinite geometric series formula on the series inside our parentheses. And this would work. However, it’s easier to simplify our series by first dividing both sides of our equation by two and then dividing both sides of our equation by 37. We see that the shared factor of 37 in the numerator and the denominator on the left-hand side of our equation cancel. Similarly, the shared factors of two in our numerator and our denominator in the right-hand side of our equation cancel. And all the factors of 37 in our numerator cancel with the factor of 37 in our denominator.
So this gives us that a half is equal to 𝑟 plus 𝑟 squared plus 𝑟 cubed. And this series keeps going. In particular, we see that this is a geometric series where the first term is 𝑟. And to get to the next term in our series, we multiply by 𝑟. We recall that the infinite sum of a geometric series with initial term 𝑎 and ratio 𝑟 is equal to 𝑎 divided by one minus 𝑟 if the absolute value of our ratio 𝑟 is less than one.
In our series, the initial term 𝑎 is equal to 𝑟, and the ratio is just equal to 𝑟. So substituting 𝑎 is equal to 𝑟 gives us the sum from 𝑛 equals zero to ∞ of 𝑟 multiplied by 𝑟 to the 𝑛th power. We see that this is equal to 𝑟 plus 𝑟 multiplied by 𝑟 plus 𝑟 multiplied by 𝑟 squared, and this keeps going. We can simplify each term in this series to see that our series is 𝑟 plus 𝑟 squared plus 𝑟 cubed, which is the same as the geometric series we were calculating. And we have that the sum of this geometric series is equal to 𝑟 divided by one minus 𝑟 if the absolute value of 𝑟 is less than one. This gives us that a half is equal to 𝑟 divided by one minus 𝑟. And it’s worth knowing that this is only true if the absolute value of 𝑟 is less than one.
We can simplify this equation by multiplying both sides by one minus 𝑟. This gives us one minus 𝑟 all divided by two is equal to 𝑟. Next, we multiply both sides of our equation by two. This gives us one minus 𝑟 is equal to two 𝑟. We add 𝑟 to both sides of our equation, giving us one is equal to three 𝑟. Then we divide both sides of the equation by three to get that one-third is equal to 𝑟.
Remember, this is only valid if the absolute value of 𝑟 was less than one. Well, the absolute value of one-third is less than one. So this is true. So we want the infinite geometric sequence with ratio of successive terms 𝑟 equal to one-third and initial value 𝑎 equal to 37. The first term in our sequence is 37. To get the second term in our sequence, we multiply 37 by a third. We then multiply 37 over three by a third to get the next term in our sequence. So that’s 37 divided by three squared, which is nine. And this pattern continues infinitely.
Therefore, we’ve shown that the infinite geometric sequence where each term in the sequence is twice the sum of the terms that follow it is equal to 37, then 37 over three, then 37 over nine, and this continues. And we keep multiplying by a third to get successive terms of this sequence.