Question Video: Finding the Sum of a Finite Geometric Sequence Mathematics

Video Transcript

Find the sum of the geometric sequence 16, negative 32, 64, and so on, all the way up to 256.

We know that the sum of any geometric sequence denoted 𝑆 sub 𝑛 is equal to 𝑎 multiplied by one minus 𝑟 to the power of 𝑛 all divided by one minus 𝑟. We can see immediately from the sequence that the value of the first term 𝑎 is 16. The second term is equal to 𝑎 multiplied by 𝑟, and this is equal to negative 32. If we label these equations one and two, we can calculate the value of 𝑟 by dividing equation two by equation one. On the left-hand side, we have 𝑎𝑟 divided by 𝑎, and on the right-hand side negative 32 divided by 16. As 𝑎 is not equal to zero, we can cancel this on the left-hand side. And negative 32 divided by 16 is negative two.

This value of 𝑟 makes sense as we multiply the first term 16 by negative two to get the second term negative 32. This also works to get from the second to third term. Negative 32 multiplied by negative two is 64. We know that the 𝑛th term of any geometric sequence written 𝑎 sub 𝑛 is equal to 𝑎 multiplied by 𝑟 to the power of 𝑛 minus one. To calculate the value of 𝑛, we can substitute our values of 𝑎 and 𝑟 and the 𝑛th term 256. This gives us the equation 256 is equal to 16 multiplied by negative two to the power of 𝑛 minus one. We can divide both sides by 16 such that 16 is equal to negative two to the power of 𝑛 minus one.

We know that negative two to the fourth power or negative 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 a value of 𝑛 equal to five. We now have values of 𝑎, 𝑟, and 𝑛. The sum of the first five terms is therefore equal to 16 multiplied by one minus negative two to the fifth power all divided by one minus negative two. This simplifies to 16 multiplied by one plus 32 all divided by three. Typing this into the calculator, we get an answer of 176. The sum of the geometric sequence 16, negative 32, 64, and so on, up to 256 is equal to 176.