# Video: Using a Given Arithmetic Sequence to Form and Solve a Linear Equation

Find π, given the arithmetic sequence (7π + 14, 4π + 4, 3π + 4).

03:26

### Video Transcript

Find π, given the arithmetic sequence seven π plus 14, four π plus four, three π plus four.

So weβve been given the first three terms of this sequence expressed in terms of an unknown π. And weβ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, π sub π, from the one that follows it, π sub π plus one, weβll always get the same value. And this will be the common difference of the sequence.

Weβve been given the first three terms of this sequence algebraically. So we can think these as π sub one, π sub two, and π sub three. And we know that if we subtract the first term from the second β so thatβs π sub two minus π sub one β weβll get the same result as if we subtract the second term from the third. Thatβs π sub three minus π sub two. Letβs now substitute the expressions weβve been given for π sub one, π sub two, and π sub three.

On the left-hand side, π sub two minus π sub one is equivalent to four π plus four minus seven π plus 14. And on the right-hand side, π sub three minus π sub two is three π plus four minus four π plus four. Now we have an equation in our unknown π. 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 π plus four minus seven π minus 14 and on the right, three π plus four minus four π minus four.

Now we simplify. On the left-hand side, four π minus seven π gives negative three π and positive four minus 14 gives negative 10. On the right-hand side, three π minus four π is negative π. And then, we have plus four minus four, which directly cancels out. So weβre left with the simplified equation negative three π minus 10 is equal to negative π. To solve, we need to collect all of the like terms on the same side. So if we add three π to each side, on the left-hand side we have negative 10 and on the right-hand side negative π plus three π gives two π.

The final step is to divide both sides of the equation by two, giving π is equal to negative five. So we found the value of π. Itβs negative five.

Itβs always good to check our answers where possible. So one way we could do this is to substitute the value of π weβ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 π plus 14, thatβs seven multiplied by negative five plus 14, which is negative 21. In the same way, the following two terms, four π plus four gives negative 16 and three π 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. Weβve found then the value of π for the given arithmetic sequence is negative five.