Tutor profile: Mitch L.
Determine whether or not the following sequence converges or diverges. If it converges, find its sum. $$ 3, 1, 1/3, 1/9, 1/27 . . .$$
To determine whether or not this series diverges, we need to determine what type of series it is. At first glance a pattern can be scene where it is being multiplied by 1/3 continuously. A constant multiplication shows that this must be a geometric series and not an arithmetic series . That ratio it is being multiplied by is 1/3, because that ratio is less than 1 and greater than -1, it must converge. To determine the value of its summation, you take the initial value and divide it by $$ 1 - ratio $$ so in this case $$ 3/(1-[1/3]) $$ this leads to an expression of $$ 3/(2/3) $$ which is simply $$ 9/2 $$ so the infinite summation of this series is $$ 9/2 $$
When x = 3 and y = 5, by how much does the value of $$ 3x^2 – 2y $$ exceed the value of $$ 2x^2 – 3y $$ ?
First, when must point in our known variables into both expression. In this question, they tell us that the variable $$ x = 3 $$ and the variable $$ y = 5 $$. To put these variables into the expression we must replace the x and y in the expression with these given values. For the first expression it would look like this $$ 3\times3^2 - 2 \times 5 $$. Using the order of operations this simplifies to $$ 27 - 10 $$ which is simply $$ 17 $$. Doing this in the second expression we get $$ 2 \times 3^2 - 3\times 5 $$. This simplifies to $$ 18 - 15 $$ or simply $$ 3 $$. To then find how much the first expression exceeds the second, we must subtract the second from the first: $$ 17 - 3 = 14 $$ So the first expression with those given x and y values exceeds the second expression by 14.
A zombie outbreak has begun in a city. At the very beginning 1 person is infected. On the first day since the outbreak 3 people are infected. The second day there are 9 people, and on the fourth day there are 81 people infected. With this information, determine what type of function best represents this rate of growth, and create a function to model this growth to the nth term.
A pattern can be seen in the number of people infected by day. At the very start, or the 0th day there was 1 single person infected. On the second day there were 3 and then 9. This is a constant multiplication by 3 to the number of people infected. The days are growing linearly, adding only 1 to each day. This linear vs multiplicative relationship is a exponential relationship. A way to find a general equation, is to determine a relationship between the variables. We know it is an exponential function, but are there any constant that are added? What is the base of the Exponential. We know that $$ B^0 = 1 $$ where B is the base. As any number taken to the 0 power equals 1, there can't be a constant. If we notice the multiplicative relationship from 1 - 3 - 9, it is being multiplied by 3 every time. So in this case, the base number will be a 3. We can write a generic equation in the form of $$ y = 3^x $$ where y is the # of people infected, and x is the day number.
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