Find the arithmetic sequence whose ninth term is negative 119, and the arithmetic mean between the third and fifth terms is negative 69.
We know that the first term of any arithmetic sequence is denoted by the letter 𝑎. And the common difference between terms is 𝑑. This means that the second term is equal to 𝑎 plus 𝑑. The third term is equal to 𝑎 plus 𝑑 plus 𝑑 or 𝑎 plus two 𝑑 and so on. Continuing this pattern, we find that the 𝑛th term is equal to 𝑎 plus 𝑛 minus one multiplied by 𝑑.
We’re told in this question that the ninth term is equal to negative 119. This means that 𝑎 plus eight 𝑑 is equal to negative 119. We’re also told that the mean of the third and fifth terms is negative 69. We can calculate the arithmetic mean of two numbers by adding them and dividing by two. Therefore, the third term plus the fifth term divided by two is equal to negative 69. Multiplying both sides of this equation by two tells us that the third term plus the fifth term is equal to negative 138. 𝑎 plus two 𝑑 plus 𝑎 plus four 𝑑 is equal to negative 138. Simplifying the left-hand side by collecting or grouping like terms gives us two 𝑎 plus six 𝑑. We can then divide both sides of this equation by two giving us 𝑎 plus three 𝑑 is equal to negative 69.
We now have two simultaneous equations that we can solve to calculate the values of 𝑎 and 𝑑. If we subtract equation two from equation one, the 𝑎’s cancel. On the right-hand side, we end up with negative 50. Subtracting negative 69 is the same as adding 69 to negative 119. Dividing both sides of this equation by five gives us 𝑑 is equal to negative 10. We can then substitute this value of 𝑑 into equation one or equation two to calculate the value of 𝑎. 𝑎 plus three multiplied by negative 10 is equal to negative 69. This can be simplified to 𝑎 minus 30 equals negative 69. Adding 30 to both sides gives us a value of 𝑎 of negative 39.
As the first term of our arithmetic sequence is negative 39 and the common difference is negative 10, then the sequence is negative 39, negative 49, negative 59, and so on.