Find the arithmetic sequence in which the sum of the first and third terms equals negative 142, and the sum of its third and fourth terms equals negative 151.
We denote the first term of any arithmetic sequence by the letter 𝑎. The common difference is denoted by the letter 𝑑. This is the difference between each of the terms in the sequence. The second term of the sequence is, therefore, equal to 𝑎 plus 𝑑. Adding 𝑑 to this gives us the third term 𝑎 plus two 𝑑. This pattern continues, giving us an 𝑛th term formula of 𝑎 plus 𝑛 minus one multiplied by 𝑑.
We are told in this question that the sum of the first and third terms is negative 142. This means that 𝑎 plus 𝑎 plus two 𝑑 is equal to negative 142. Simplifying this gives us two 𝑎 plus two 𝑑 equals negative 142. We could divide both sides of this equation by two. However, we’ll work out the second equation in this question first.
The third and fourth terms have a sum of negative 151. This means that 𝑎 plus two 𝑑 plus 𝑎 plus three 𝑑 is equal to negative 151. Grouping or collecting the like terms gives us two 𝑎 plus five 𝑑 is equal to negative 151. We now have two simultaneous equations that we can solve to calculate the values of 𝑎 and 𝑑.
When we subtract equation one from equation two, the 𝑎’s cancel. Five 𝑑 minus two 𝑑 is equal to three 𝑑. Subtracting negative 142 is the same as adding 142 to negative 151. This is equal to negative nine. Dividing both sides of this equation by three gives us 𝑑 is equal to negative three. We now need to substitute this value into equation one or equation two to calculate 𝑎.
Substituting into equation one gives us two 𝑎 plus two multiplied by negative three is equal to negative 142. This simplifies to two 𝑎 minus six is equal to negative 142. Adding six to both sides gives us two 𝑎 is equal to negative 136. And finally, dividing by two gives us 𝑎 equals negative 68. As the first term of the sequence is negative 68 and the common difference is negative three, then the arithmetic sequence is negative 68, negative 71, negative 74, and so on.