Find the value of the first negative term in an arithmetic sequence in which the 27th term minus the 49th term is equal to 44 and the second term plus the 33rd term equals 70.
In an arithmetic sequence, we identify the first term as 𝑎 sub one. And to get from the first term to the second term, we have a common difference of 𝑑. To get to the third term, we take the second term and add the common difference to it. Each term is the previous term plus the common difference. We write 𝑎 sub 𝑛 to identify the 𝑛th term in the sequence. We can define any term in the sequence 𝑎 sub 𝑛 with respect to two things: the first term and the common difference. This means, in order for us to solve for the first negative term in this sequence, we’ll need to know two things: the first term and the common difference.
However, we haven’t been given either of those things, which means before we can solve for the first negative term, we need to find the first term and the common difference. We can do that by rewriting the two equations we were given with respect to their first term and to the common difference. We know that the 27th term will be equal to the first term plus 26 times the common difference. We also know that the 49th term will be equal to the first term plus 48 times the common difference. If we take these two new equations and substitute them in for the 27th and 49th term, we’ll have 𝑎 sub one plus 26𝑑 minus 𝑎 sub one plus 48𝑑 equals 44.
To solve this, we need to be really careful to distribute the subtraction across the 𝑎 sub one and the 48𝑑. When we do that, we see that we have 𝑎 sub one minus 𝑎 sub one, and those cancel out. And then we have 26𝑑 minus 48𝑑, which is negative 22𝑑. So we have negative 22𝑑 equals 44, and we can solve for 𝑑 by dividing both sides of the equation by negative 22, which tells us that 𝑑 equals negative two. Now, we have one of the pieces of information we need. We know the common difference. But we’ll still need to use the second equation to solve for the first term. We know that 𝑎 sub two will be equal to the first term plus the common difference. And the 33rd term will be equal to the first term plus 32 times the common difference.
We’ll substitute these values in for the second and 33rd term, combine our like terms. Two 𝑎 sub one plus 33𝑑 equals 70. And at this point, we’re able to substitute in what we know about 𝑑 because we know that 𝑑 equals negative two. So two 𝑎 sub one minus 66 equals 70. Add 66 to both sides of the equation, and two 𝑎 sub one equals 136. Divide both sides by two, and we find our first term is 68. If our first term is 68 and the common difference is negative two, our second term will be 66 and our third term will be 64.
We want to find the value of the first negative term. If we look at this sequence, we see that we have 68, 66, 64. And that means that this sequence is even numbers. The first negative term will be the first negative even number, negative two. However, if you didn’t immediately recognize that pattern, there is something else we can do. We know that 𝑎 sub 𝑛 equals 𝑎 sub one plus 𝑛 minus one 𝑑. We know that our first term is 68, and our common difference is negative two. We can distribute this negative two and say that our 𝑎 sub 𝑛 equals 68 minus two 𝑛 plus two. And combining 68 plus two, we get 70. In this term, any 𝑎 sub 𝑛 is found by taking 70 and subtracting two 𝑛.
But we want to know when this equation is less than zero. When is 70 minus two 𝑛 less than zero? Add two 𝑛 to both sides, and we have 70 is less than two 𝑛. Divide both sides by two. And we see that 35 has to be less than 𝑛 or 𝑛 greater than 35. Now, think about what the 𝑛 means in this context. It’s telling us that the term number must be greater than 35. Our term numbers are all integers, so the integer that comes next and is greater than 35 would be 36. And if we’ve done our calculations correctly, the 36th term should be the first negative term in the sequence. Remember, we find our term by 70 minus two 𝑛. We’re looking at the 36th term, 𝑛 equals 36. When we solve this, we see that the 36th term is negative two.
If we wanted to check and make sure this was exactly the first negative term, we can see that the 35th term is equal to 70 minus two times 35. The 35th term is zero. The 36th term is the first negative term. But our question is asking for the value of the first negative term not which term it is. And that makes our final answer negative two.