Find 𝑎, given the arithmetic sequence seven 𝑎 plus 14, four 𝑎 plus four, three 𝑎 plus four.
So we’ve been given the first three terms of this sequence expressed in terms of an unknown 𝑎. And we’re told that this sequence is arithmetic. We know that in an arithmetic sequence, the difference between consecutive terms is always the same. This means that if we subtract any term, 𝑎 sub 𝑛, from the one that follows it, 𝑎 sub 𝑛 plus one, we’ll always get the same value. And this will be the common difference of the sequence.
We’ve been given the first three terms of this sequence algebraically. So we can think these as 𝑎 sub one, 𝑎 sub two, and 𝑎 sub three. And we know that if we subtract the first term from the second — so that’s 𝑎 sub two minus 𝑎 sub one — we’ll get the same result as if we subtract the second term from the third. That’s 𝑎 sub three minus 𝑎 sub two. Let’s now substitute the expressions we’ve been given for 𝑎 sub one, 𝑎 sub two, and 𝑎 sub three.
On the left-hand side, 𝑎 sub two minus 𝑎 sub one is equivalent to four 𝑎 plus four minus seven 𝑎 plus 14. And on the right-hand side, 𝑎 sub three minus 𝑎 sub two is three 𝑎 plus four minus four 𝑎 plus four. Now we have an equation in our unknown 𝑎. So we need to solve it. The first step will be on each side we need to distribute the negative sign over the set of parentheses. On the left-hand side, we now have four 𝑎 plus four minus seven 𝑎 minus 14 and on the right, three 𝑎 plus four minus four 𝑎 minus four.
Now we simplify. On the left-hand side, four 𝑎 minus seven 𝑎 gives negative three 𝑎 and positive four minus 14 gives negative 10. On the right-hand side, three 𝑎 minus four 𝑎 is negative 𝑎. And then, we have plus four minus four, which directly cancels out. So we’re left with the simplified equation negative three 𝑎 minus 10 is equal to negative 𝑎. To solve, we need to collect all of the like terms on the same side. So if we add three 𝑎 to each side, on the left-hand side we have negative 10 and on the right-hand side negative 𝑎 plus three 𝑎 gives two 𝑎.
The final step is to divide both sides of the equation by two, giving 𝑎 is equal to negative five. So we found the value of 𝑎. It’s negative five.
It’s always good to check our answers where possible. So one way we could do this is to substitute the value of 𝑎 we’ve calculated into each term in the sequence and then check that the three values do indeed have a common difference. For the first term, seven 𝑎 plus 14, that’s seven multiplied by negative five plus 14, which is negative 21. In the same way, the following two terms, four 𝑎 plus four gives negative 16 and three 𝑎 plus four gives negative 11. These three terms have a common difference of positive five. Negative 21 plus five gives negative 16. Negative 16 plus five gives negative 11. So the sequence is indeed arithmetic. We’ve found then the value of 𝑎 for the given arithmetic sequence is negative five.