The first 3 terms of a geometric sequence are, respectively, the 1st, 4th and 13th terms of an arithmetic sequence.
Given that the first term of the geometric sequence is 3 and the common difference of the arithmetic sequence is 2, calculate the sum of the first 4 terms of the geometric sequence.
What I did was determine the 1st, 4th and 13th terms in the arithmetic sequence to be 3, 9 and 27. I then determined the common ratio for the geo sequence to be 3. This means that the 4th term in the geo sequence is 81. The sum of these 4 terms is 120. I am not 100% confident in my working, so if anyone could confirm whether I have done this correctly, it would be greatly appreciated.
Best Answer
Your reasoning is exactly right. The formula for the $n$th of a geometric sequence is $ar^n$ where $a$ and $r$ are arbitrary constants.
In this case (if we start our numbering at one instead of zero) $a$ is 1 and $r$ is 3. So the terms are the powers of three: 3, 9, 27, 81.
(If you start your numbering at zero, then $a$ is 3 and $r$ is also 3, for the same result. Which version to use depends on your instructor, since they're mathematically equivalent. I find the zero-based one more intuitive, but I'm also a computer scientist more than a mathematician, so take that with a grain of salt.)