Find 𝑥 given three consecutive terms of an arithmetic sequence are negative 10𝑥, negative four 𝑥 minus two, and negative 𝑥 plus eight.
In this question, we’re given three consecutive terms of an arithmetic sequence: negative 10𝑥, negative four 𝑥 minus two, and negative 𝑥 plus eight. And we need to use this information to find the value of 𝑥. To do this, let’s start by recalling exactly what we mean by an arithmetic sequence. We recall, in an arithmetic sequence, the difference between any two consecutive terms must always be the same. And we can see that this is useful because we’re actually given three consecutive terms in our arithmetic sequence in the question.
Therefore, because this is an arithmetic sequence, the difference between any two of these consecutive terms has to be the same. This is also called the common difference of our arithmetic sequence. And one way we could try and find the common difference of this arithmetic sequence is to find the difference between the second term and the first term of our sequence. This is negative four 𝑥 minus two, and then we subtract negative 10𝑥.
However, this is not the only way we could calculate the common difference of this arithmetic sequence. We could’ve also found the difference between the third term in our sequence and the second term in our sequence. That’s negative 𝑥 plus eight minus negative four 𝑥 minus two. And both of these expressions are equal to the common difference of our expression, so they must be equal. This comes directly from the definition of an arithmetic sequence. The difference between any two consecutive terms in our sequence has to be the same.
So let’s try evaluating this equation. On the left-hand side of this equation, we’re subtracting negative 10𝑥. And we know this is the same as adding 10𝑥. And on the right-hand side of this equation, we’re going to want to distribute the negative over our parentheses. This gives us negative 𝑥 plus eight plus four 𝑥 plus two, because a negative times a negative is a positive.
Now, we want to collect like terms. On the left-hand side of our equation, we have negative four 𝑥 plus 10𝑥, which is equal to six 𝑥. And on the right-hand side of our equation, we have negative 𝑥 plus four 𝑥, which is equal to three 𝑥, and eight plus two, which is equal to 10. So this gives us six 𝑥 minus two is equal to three 𝑥 plus 10.
But remember, the question is trying to make us find the value of 𝑥. So we need to rearrange this equation to make 𝑥 the subject. We’ll do this by subtracting three 𝑥 from both sides and adding two to both sides of our equation. This then gives us that three 𝑥 is equal to 12. And we can find the value of 𝑥 by dividing through by three. We get that 𝑥 is equal to 12 divided by three, which we know is equal to four, which is our final answer.
Therefore, we were able to show if the sequence negative 10𝑥, negative four 𝑥 minus two, and negative 𝑥 plus eight are three consecutive terms in an arithmetic sequence, then the value of 𝑥 has to be equal to four.