Video Transcript
By finding the sum of an infinite geometric sequence, express negative 2.7 recurring as a mixed number.
The key to this question is the part that’s circled; it’s the seven with the line above it, which means seven recurring. Let’s begin by recalling what we mean by a recurring number or recurring decimal.
There are a couple of common ways of writing a recurring decimal, firstly, as in this case, with a line above the digit. Alternatively, we write a dot above the digit or digits that are recurring or repeating. In both of these cases 0.7 recurring means the number 0.7777 and so on. The number seven would repeat indefinitely.
There are several ways of rewriting a recurring decimal as a fraction. And in this question, we’re asked to do so using our knowledge of geometric sequences. We begin by rewriting 0.7 recurring as 0.7 plus 0.07 plus 0.007 and so on. The right-hand side of our equation is a geometric series with first term 𝑎, second term 𝑎𝑟, third term 𝑎𝑟 squared, and so on.
The terms of the series could be rewritten as a geometric sequence with first term 𝑎 equal to 0.7. We could calculate the common ratio 𝑟 by dividing any term by the preceding one. For example, 0.07 divided by 0.7 is equal to 0.1. Next, we recall that if the absolute value of the common ratio is less than one, the sum to ∞ of a geometric sequence is equal to 𝑎 divided by one minus 𝑟. The sum to ∞ of our sequence is therefore equal to 0.7 divided by one minus 0.1. This is equal to 0.7 over 0.9, which simplifies to seven-ninths.
This means that 0.7 recurring is equal to seven-ninths. In this question, we are trying to express negative 2.7 recurring as a mixed number. This is therefore equal to negative two and seven-ninths.
