The first 4 terms of a sequence are {3,23,33,43,…}\left\lbrace\sqrt3,2\sqrt3,3\sqrt3,4\sqrt3,\ldots\right\rbrace{3,23,33,43,…}. Continuing this pattern, find the 7th⁡7^{\th}7th term.

7 3 7\sqrt3 7 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

The first 4 terms of a sequence are { 3 , 2 3 , 3 3 , 4 3 , … } \left\lbrace\sqrt3,2\sqrt3,3\sqrt3,4\sqrt3,\ldots\right\rbrace { 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> , 2 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> , 3 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> , 4 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> , … } . Continuing this pattern, find the 7 th ⁡ 7^{\th} 7 t h term.

Everyone. Welcome back. So in this problem, we have the first four terms of a sequence that are given to us. We've got radical 3, 2 radical 3, 3 radical 3, 4 radical 3, and so on. This is an infinite sequence because it keeps on going forever. And continuing this pattern, we wanna find out what the 7th term of the sequence is. Let's go ahead and get started here. Now remember, each one of these numbers that are separated by commas are called the terms of a sequence. And the way that we use the notation for this is we have that a with a little subscript just means the index or that term. So for example, a one is the first term in the sequence, and in this sequence, it's just radical 3. Notice how a 2 is 2 radical 3. A 3 is 3 radical 3. We wanna use this to eventually get to what the 7th term in the sequence is. That would be a 7. Alright. So let's go ahead and look at this pattern here. Hopefully, by now, you kinda start to see that there's a little bit of a pattern emerging. I'm gonna continue with a 4. A 4 would be 4 radical 3. All of these, I really just pulled from this sequence over here that that was given to me. Notice how it happens is that the index, that number that's at the subscript of the a, is the same number that goes in front of the radical 3. So for a 4, it goes with 4. This 3 over here means that there's a 3 in front of the radical 3. This 2 means that there's a 2 in front of the radical 3. Even this one means that there's a sort of a hidden one in front of this radical 3. So whatever the index is, that's the number that goes in front of the radical 3. The radical 3 never changes. It's always the same. So if you wanna continue that pattern, what would the 7th number be? It would be a 7. Well, you would just do the same thing. It would just be a 7, so that means that there would be a 7 in front of the radical 3. So this would be the 7th term in the sequence, which is b 7 radical 3. That's the answer. Now the problem doesn't ask you to do this, but if you wanted to generalize this as a formula, this would basically just be that a n, whatever the nth term that you wanted to find, would just be n times radical 3. Right? The index, whatever goes here, is whatever number that goes in front of the radical 3. So if you wanted to figure out the 100th term, it would just be 100 times radical 3. So this would be the sort of the general formula to find any one of these terms. That's it for this one, folks. Let's keep going.