Find the sum of the first six terms of the geometric sequence beginning 405, 135, and 45.
The sum of the first 𝑛 terms of any geometric sequence is given by the formula 𝑆 of 𝑛 is equal to 𝑎 multiplied by one minus 𝑟 to the power of 𝑛 divided by one minus 𝑟, where 𝑎 is the first term, 𝑟 is the common ratio, and 𝑛 is the number of terms.
In our example, the first term is 405. Therefore, 𝑎 is equal to 405. The common ratio 𝑟 is equal to a third as 405 multiplied by a third is equal to 135. Likewise, 135 multiplied by a third is equal to 45. To get from the first term to the second and from the second to the third, we need to divide by three or multiply by one-third.
We were asked to find the sum of the first six terms. Therefore, 𝑛 is equal to six. Substituting these values into the formula gives us 405 multiplied by one minus one-third to the power of six divided by one minus one-third.
Typing the numerator into our calculator gives us 404.4 recurring or three thousand, six hundred and forty-ninths. The denominator one minus a third is equal to two-thirds or 0.6 recurring. Typing this into the calculator gives us an answer of 606.6 recurring. This can also be written as one thousand, eight hundred and twenty thirds or 1820 over three.
The sum of the first six terms of a geometric sequence with first term 405 and common ratio one-third is 606.6 recurring or 1820 divided by three.