In a geometric sequence, the first term is 𝑎, the common ratio is 𝑟, and the last term is 𝑙. Find the sum of the geometric sequence with 𝑎 equal to 1408, 𝑟 equal to one-half, and 𝑙 equal to 88.
We recall two formulae that we can use when dealing with geometric sequences. Firstly, the 𝑛th term 𝑎 sub 𝑛 is equal to 𝑎 multiplied by 𝑟 to the power of 𝑛 minus one. Secondly, the sum of the first 𝑛 terms 𝑆 sub 𝑛 is equal to 𝑎 multiplied by one minus 𝑟 to the power of 𝑛 all divided by one minus 𝑟. We are told in the question that 𝑙 is equal to 88. As this is the last term, this will be equal to 𝑎 sub 𝑛. Substituting in our values of 𝑎 and 𝑟, we have the equation 88 is equal to 1408 multiplied by one-half to the power of 𝑛 minus one.
Dividing both sides of the equation by 1408, we get one sixteenth is equal to a half to the power of 𝑛 minus one. We might recognize here that one-half to the power of four is equal to one sixteenth. We know this as when we raise any fraction to a power, we can raise the numerator and denominator to the power separately. In this case, one to the power of four is equal to one and two to the power of four is equal to 16. This means that 𝑛 minus one must be equal to four. Adding one to both sides of this equation gives us 𝑛 is equal to five. There are five terms in the geometric sequence.
If we didn’t recognize that a half to the fourth power was equal to one sixteenth, we could have used logarithms. We know that if 𝑎 is equal to 𝑏 to the power of 𝑥, then 𝑥 is equal to log base 𝑏 of 𝑎. This means that, in our question, 𝑛 minus one is equal to log to the base half of one sixteenth. This is equal to four. So once again, 𝑛 minus one is equal to four, so 𝑛 is equal to five. We can now calculate the sum of the five terms in the geometric sequence. This is equal to 1408 multiplied by one minus one-half to the fifth power all divided by one minus a half. Typing this into the calculator, we get 2728.
The sum of the geometric sequence with 𝑎 equal to 1408, 𝑟 equal to one-half, and 𝑙 equal to 88 is 2728.