00:01
Consider the geometric sequence where a n equal 3 fourth to the nth power for where n is a positive integer number, that is, n equal 1, 2, 3, and so on and so forth.
00:14
In part a, which terms are less than 0 .0001? part b, what is the sum of the first 10 terms? terms, we'll see how much is s equal the series from n equal 1 to infinity of a n, that is the sum of infinitely many terms, and in part d we solve for big n, the inequality absolute value of s, the sum of the infinitely many terms, minus the sum of the terms from n equal 1 to big n, and that absolute value less than 0 .0001 that is indeed the error of the series that is a partial sum up to n minus the value of the infinity many the sum of the infinity many terms good so in part a what we want to know is which terms of the sequence a n which terms of that sequence are less than 0 .0001 that is 10 to the negative 4 so we want to know when a n is less than 10 to the negative 4 n is clear that all terms in this sequence are positive numbers because the base 3 -fold is a positive number and we will erase it to a positive inter power we are just multiplying that positive number by itself with number of times equal to the exponent so the result is always a positive number so we want to know specifically which are the indices of the sequence for which this inequality is true so we see that this inequality you want to solve a n less than 10 to negative 4 is the same as or is equivalent to log base 10 of a n less than log base 10 of 10 to an 84 that's because the logarithmic function is increasing an increasing function is very important to remember is always it's also only defined for the positive number but we have explained that all these terms a n are positive numbers so we can apply the logarithm to those terms and because the function is increasing the inequality is preserved and also we go back that is we can talk about these two things being equivalent because the exponential function 10 to to the x power is also increasing function so we can go back applying the tenths to the x function which is the inverse of the logarithm base 10 and we get this inequality here okay this is equivalent to that and we say this is equivalent to logarithm base 10 of three fourth to the nth power that's just the definition of a n here is geometric sequence and that less to less than no logarithm base 10 of 10 to negative 4 because we have a power of 10 and we are taking logarithm base 10 the result of this logarithm on the right is negative 4 so what we get now is applying the properties of logarithm we have a power inside the argument of the logarithm on the left side of the inequality, so this exponent goes down, multiplying the logarithm by 10 of the base 3 -fold, and that is less than negative 4, and now we want to divide both sides by the logarithm by 10 of 3 -fold, but remember the logarithm is negative when the argument is between 0 and 1.
04:56
That's the case of 3 fourths so this is equivalent to n greater than negative 4 over logarithm base 10 of 3 fourths that is because we are dividing both sides of this inequality by logarithm base 10 of 3 fourths which is negative so here we get to explain that since logarithm base 10 of 3 3 fourths is negative because 3 fourths is positive and less than 1.
05:37
The logarithm in any base of a number between 0 and 1 is negative.
05:43
And because we are dividing both sides of this inequality here by a negative number, we got to shift the inequality from less than to greater than.
05:55
So n greater than negative 4 divided by logarithm by the state of 3 fourths.
06:01
So logarithm base 10 of 3, 4 being negative, when we divide negative 4 by a negative number, we get a positive number, and it makes sense.
06:11
Okay, so now this number is going to be calculated using a calculator, and you can verify this is about 32 .02.
06:26
That is, n got to be greater than 32 .02.
06:31
That is, we cannot say 32 will be a solution because n being greater than 32 .02, then 32 is not included.
06:43
So we start having the inequality, this equivalence, give us this result when n starts its values at 33.
06:56
Let me put that here, so n greater than or equal to 33.
07:01
So that's the answer, because the answer will be then that the terms of the sequence are less than 0 .0001 when the index of the term of the sequence is greater than or equal to 33.
07:20
If you take 32, you can verify that when you calculate 3 -folds raised to the power 32, you don't have the inequality right here.
07:32
You don't have that the resulting term is less than 10 to the 94th.
07:37
That starts to happen when n is 33 and, of course, when it's greater than 33 also.
07:46
So that's the answer in part a.
07:48
So let me put this part a okay.
07:50
Okay, now part b, in part b we need to use, and then in the other next part, we get to use a formula to calculate the sum of the terms of the sequence from n equal 1 up to a number.
08:04
So i'm going to write down that formula we're going to use several times.
08:09
So we have the, let's say, let's call s n or s big n, better...