Video: Solving Word Problems Involving Geometric Series

On the first day, there were 810 bacteria on a petri dish. Find, giving the answer to the nearest integer, the bacterial count after six days given the bacteria replicate daily.

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Video Transcript

On the first day, there were 810 bacteria on a petri dish. Find, giving the answer to the nearest integer, the bacterial count after six days given the bacteria replicate daily.

So in order to find a count, that means we’re finding the amount that’s there. So we’re adding everyday how much there actually is, and that’s called a summation. And this one is going to go on for six days. The formula for this summation will be: 𝑎 times one minus 𝑟 to the 𝑛 power divided by one minus 𝑟.

So 𝑟 will represent our rate and it replicates daily. So it’s multiplying by two. So on the first day there might be one and on the next day there should be another one, so it’s twice as much. And then after that, the next day if it’s going to replicate what it already has, then you’re gonna add two. So now you have four. And then if you’re gonna replicate how much that is, you’re gonna add four again. So essentially, you’re just taking whatever number that it is and multiply it by two.

𝑎 represents the initial amount that we have and that would be 810. And 𝑛 will represent the number of days how many times this is actually happening, and they’re six days; so 𝑛 is six.

So we have 810 times one minus two the sixth power divided by one minus two. Two to the sixth power, 64 and one minus two is negative one. One minus 64 is negative 63. So now let’s take 810 times negative 63 and then divide by negative one, and we get 51030.

Therefore, after six days, there will be 51030 bacteria in the petri dish.

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