The sequences in Exercises 13–18 are defined using recursion form...

Question

The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a_1=3 and a_n=4a_n-1 for n≥2

Accepted Answer

Hello, everyone in this video, we're gonna be looking at this practice problem where we want to write the first four terms of a sequence that uses a recursion formula where A one is equal to five and A N is equal to seven A N minus one for and greater than, or equal to two. So basically starting from the second term and above, we have to use this formula A N is equal to seven times A and minus one to solve fourth terms. And because we're trying to solve for the first four terms, we need to solve for N equals one through N equals four. So let's start with N equals one that is a one or the first term of the A sequence. And we're actually given the value it is five. So if we go to an equals two, we now have this recursion formula that we need to use where we have A and so a two is equal to seven times A two minus one. I start to simplify this, I get seven times a one and we know the value of a one because it's given to us. So I can plug that in. So I have seven times a one which is five. So a two is equal to seven times five, which is 35. So now I have a value for my second term. So solving for my third term where N is equal to three, I have a three is equal to seven, a three minus one. If I start to simplify that I get seven times A two and we know the value of a two because we just stall for it. So I get seven times 35. And that means that a three is equal to 245. So now, finally, my fourth term where N is equal to four. Again, I need to use the recursion formula where A four is equal to seven times a four minus one or seven times a three. And again, I just solved for a three. So I can plug that in and this becomes seven times 2 45 which means that a four is equal to 1715. And here you can see that. Now I have the first four terms of the sequence so I can pull them out. The first one is five, the second one is 35. The third is 245 and the fourth is 1715. So this becomes my final answer for this question. And you can see that this aligns with answer choice D. So hopefully, this video helped and see you next time.

The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a_1=3 and a_n=4a_n-1 for n≥2

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