And on the right, we get □ to the power of □ minus one times □ or □ to the power of one. On the left-hand side, when we do, we get □ over 81, nothing groundbreaking. And we’ll see why we do that in a moment. So we’re going to do something a little bit strange with this first equation. And now we have an equation in terms of □ to the power of □ minus one and then one that has □ to the power of □ and □. And on the right, we get 324 minus 324 times □ to the □th power. On the left-hand side, multiplying both one and negative □ by 484 gives us 484 minus 484□. Let’s distribute each pair of parentheses. So we have 484 times one minus □ equals 324 times one minus □ to the □th power. And that gets rid of the fraction for us. We’re going to multiply through by one minus □ here. So we say that 484 is equal to 324 times one minus □ to the □th power all over one minus □. In our other formula, we know that the sum of the first □ terms is 484. So we find that □ to the power of □ minus one is equal to one over 81. Our first formula, the one for the □th term, becomes four equals 324 times □ to the power of □ minus one. Let’s substitute everything we have into each of our formulae. And then, of course, we know the sum of all our terms to be equal to 484. So we can write □ sub □ to be equal to four. But we don’t actually know how many terms are in our sequence. Now we’re told the first term in our sequence is 324. And so we recall the sum of the first □ terms of the geometric sequence, with a first term □ sub one and a common ratio □, is □ sub one times one minus □ to the □th power over one minus □. Similarly, we’ve been given some information about the sum of the terms in the sequence. It’s the number we multiply each term by to get the next term. The □th term of a geometric sequence, □ sub □, is given by □ one times □ to the power of □ minus one, where □ sub one is the first term in the sequence and □ is the common ratio. So let’s recall what we do know about a geometric sequence. But we’re dealing with a geometric sequence. Find the geometric sequence, given the first term is 324, the last term is four, and the sum of all the terms is 484.Īt first glance, it might feel like we don’t have enough information to answer this question.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |