### Video Transcript

Part a) The πth term of a sequence is three π plus three to the power of two π. Work out the fourth term of the sequence.

The πth term is just a way of describing the general term in a sequence. It tells us what the term will be equal to when the term number is equal to π. For example, to find the first term in a sequence, we substitute π equals one. To find the second, we substitute π equals two, and so on. In this question, weβre asked to work out the fourth term of the sequence. So weβre going to substitute π equals four into the general term.

First, we have three π, which means three times π. So this is equal to three times four. Then weβre adding three to the power of two π. So this is three to the power of two times four. Now three times four is just 12. And for the second term, in the power we have two times four, which is eight. So this simplifies to 12 plus three to the power of eight. Now three to the power of eight just means three multiplied together eight times, but we donβt need to type this into our calculator. We can use the power button instead. What this button looks like will vary depending on your calculator. But on my calculator, itβs the button with an π₯ and then a little box where the power would go.

So if I type three and then press this button and then eight and then press equals, it evaluates three to the power of eight for me. Three to the power of eight is equal to 6561, which you can confirm by doing the calculation longhand if you wish. Finally, adding the 12 then gives 6573. So this is the fourth term of the sequence with πth term three π plus three to the power of two π.

Part b) The πth term of another sequence is three π multiplied by three to the power of two π. Circle an equivalent expression for the πth term. The options are three to the power of two π plus one multiplied by π, nine to the power of two π plus three π to the power of two π, nine π to the power of three π, or nine to power of two π multiplied by π.

To answer this question, we can just break this product down slightly differently. So three π means three multiplied by π. And as it doesnβt matter what order we multiply things together in, we can move this π to the end of the expression. So we have three multiplied by three to the power of two π multiplied by π. We can see then that at the start of this expression we have three multiplied by three to the power of two π. And we need to think about how to simplify this. To do this, weβll need to use our laws of indices.

Now, there isnβt a power written for three, but in fact three is just equal to three to the power of one. So we can include this power if we wish. One of our laws of indices tells us that if weβre multiplying together two terms with the same base β that means the letter or number thatβs being raised to the power, so in the rule Iβve written thatβs π₯, but in the question itβs three β then we can add together the powers. So the general rule is that π₯ to the power of π multiplied by π₯ to the power of π is equal to π₯ to the power of π plus π.

So three to the power of one multiplied by three to the power of two π can be written as three to the power of the sum of those powers. Thatβs two π plus one. Remember, weβre still multiplying by π. But in algebra, we donβt need the multiplication sign between the two things weβre multiplying, so we can eliminate it. Weβre left with three to the power of two π plus one multiplied by π or three to the power of two π plus one π, which is the same as the first option we were given for an equivalent expression. So our answer then is that an equivalent expression for the πth term of this sequence is three to the power of two π plus one π.