Find the first three terms of a geometric sequence given that 𝑎 six, so the sixth term, is equal to negative 3616 and the common ratio is negative two.
As we know that the sequence we’re looking at is a geometric sequence, we know then that we can actually use a formula to find any term in that sequence, and this is that formula; it says that 𝑟, value of our term, is equal to 𝑎 multiplied by 𝑟 to the power of 𝑛 minus one, where 𝑎 is our first term, 𝑟 is our common ratio, and 𝑛 is our term number.
Okay, great! So now, we can use this to actually find what our first term is. So we can say as we know the sixth term 𝑎 six, we can say that the sixth term is equal to 𝑎 multiplied by 𝑟 to the power of five. And it’s to the power of five because our term number six minus one. Now, we can substitute in our values so that we can calculate 𝑎. So we get negative 3616 is equal to 𝑎 multiplied by negative two to the power of five.
So therefore, we get negative 3616 is equal to negative 32 𝑎. Then to find our 𝑎, we’re just gonna divide both sides by negative 32. And therefore we know that our first term 𝑎 is equal to 113.
Great! So we can now use this to find term two and term three. Well, as we now have the common ratio and the first term and we’re actually finding three consecutive terms, all we need to do is to find the next term, is to multiply the previous term by a common ratio.
So to find our second term, we’re gonna multiply our first term, which is 𝑎, 113, by our common ratio, negative two, which gives us the second term value of negative 226.
Then, finally, to find our third term, we’ll have negative 226, which is our second term, multiplied by negative two, our common ratio, which gives us the third term value of 452. So there we have it. We’ve now found the first three terms of the geometric sequence.