Find the seven-term arithmetic sequence whose middle term is negative 105 and the sum of the last three terms is negative 369.
First, let’s think about this seven-term sequence. We can list out the terms like this where 𝑎 sub one is the first term, 𝑎 sub two is the second term, all the way up until 𝑎 sub seven for the seventh term. This type of sequence is identified with a common difference of 𝑑, a value that’s added to the first term to give us the second term. That same amount is added to each term. It means we have a formula for finding the term 𝑛. You take the first term and you add to it 𝑛 minus one times 𝑑. For example, if we were looking for the fourth term, we would start with the first term. Since we’re looking for the fourth term, 𝑛 minus one is four minus one. That’s three times 𝑑.
We can see from the first term to the fourth term three 𝑑s are added. Now, let’s start by identifying what we know about this specific sequence. We know that it has seven terms and that its middle term is negative 105. Out of seven terms, the middle is four. In our case, the middle term 𝑎 sub four is negative 105. So we can go ahead and add that to our line. We also know that the sum of the last three terms is negative 369. The last three terms are 𝑎 sub five, 𝑎 sub six, and a sub seven. And if we add the fifth, sixth, and seventh terms together, we’ll get negative 369.
Now, we know what the fourth term is. And we also know that the fifth term is going to be equal to the fourth term plus the common difference. The fourth term is negative 105. Negative 105 plus 𝑑 equals the fifth term. And the sixth term will be equal to negative 105 plus 𝑑 plus 𝑑. So we can write the sixth term as negative 105 plus two 𝑑. Following the same pattern, we can write our seventh term as negative 105 plus three 𝑑. And all of this will be equal to negative 369.
If we combine the like terms, we have negative 105 plus negative 105 plus negative 105, negative 315. 𝑑 plus two 𝑑 plus three 𝑑 equal 60. So we’ll have negative 315 plus 60 is equal to negative 369. And then we can add 315 to both sides, which will give us 60 is equal to negative 54, then divide both sides by six. And we’ll find that 𝑑, our common difference, is negative nine. We can use this negative nine to find the other sixth terms. So we plug in negative nine everywhere we found the common difference. And to get our fifth term in the sequence, we’ll take negative 105, the fourth term, and subtract nine, which gives us negative 114.
From the fifth term, we subtract the common difference of negative nine, which gives us the sixth term of negative 123. And then we subtract nine from negative 123. We get negative 132. To find the third term, we need to ask what minus nine is equal to negative 105. And to find that, we’ll add nine to negative 105 which is negative 96. Negative 96 minus nine equals negative 105. And our second term will be negative 96 plus nine, which is negative 87. And finally, our first term negative 87 plus nine equals negative 78. So we can say this seven-term sequence begins with negative 78 has a common difference of negative nine in the seventh term would be negative 132.