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Genrish500 [490]
3 years ago
12

what is the first term of a geometric sequence in which the second term is 6 and the fourth term is 54

Mathematics
1 answer:
IRISSAK [1]3 years ago
5 0
For a geometric sequence:
U_{n} = a * r^{n - 1}
U_{n} = the nth term, i.e. U₂ is the second term, U₃ is the third term, etc.
a = first term (U₁)
r = multiplier
n = number corresponding to the postion of the nth term in the sequence, e.g. for U₂, n = 2, for U₃, n = 3

According to the information:
U₂ = 6
U₄ = 54
So:
U_{2} = a * r^{(2) - 1} = 6 \\\\ a * r^{1} = 6 \\\\ ar = 6 \\
U_{4} = a * r^{(4) - 1} = 54 \\\\ a * r^{3} = 54 \\\\ ar^{3} = 54
Now, we have two equations in terms of a and r;
We can work out a by first working out r;
If we let the U₂ eq'n be [1] and the U₄ eq'n be [2];
Then, we do [2] ÷ [1] to cancel out the a and get an equation with only r that we are able to solve and then just solve it, like so:
\frac{ar^{3}}{ar} =  \frac{54}{6} \\\\ r^{2} = 9 \\\\ r = 3
Note: r has to be 3, and not -3, as the two given terms of the sequence tell us that the terms of the sequence will get bigger from each term to the next so r has to be above 1.

Now that we know r, we can sub it back into either [1] or [2] and solve for a, Its quicker and easier to sub it into [1] as we have no powers greater than 1 and we have smaller numbers to deal with (but you would get the same value of a either way), so:
U₂ = ar = 6
a(3) = 6
a = 6/3
a = 2
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Guide the discussion so that students understand that a ratio is simply the quotient of two numbers, while a rate is the ratio of two measurements that have different units (like miles and hours, or dollars and ounces). Reinforce the fact that a rate is usually expressed in per unit form, where the denominator is 1.

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5 0
3 years ago
Question 3: In les Miserables Monsier, Thenardier, and Madame Thenardier are married and during act two, Cosette and Marius beco
Natasha2012 [34]
Answer: 14,515,200

Note: this is a single number (not an ordered triple or a collection of three different numbers) roughly equal to about 14.5 million if you round to the nearest hundred thousand.

----------------------------------------------

Explanation:

There are 13 people. Let's call them person A, person B, person C, ... all the way up to person M. The first four people are given who we'll call A through D. The rest (E through M) aren't really important since they aren't named. 

A = Monsier Thenardier
B = Madame Thenardier
C = Cosette
D = Marius
Peron's E through M = remaining 9 people

------------------

A and B must stick together. Because of this, we can consider "AB" as one "person".
So we go from 13 people to 13-2+1 = 12 "people". 

Likewise, C and D must stick together. We can consider "CD" as one "person". So we go from 12 "people" to 12-2+1 = 11 "people"

------------------

The question is now: how many ways can we arrange these 11 "people" around a circular table? The answer is (n-1)! ways where n = 11 in this case

So, (n-1)! = (11-1)! = 10! = 10*9*8*7*6*5*4*3*2*1 = 3,628,800

We're almost at the answer. We need to do two adjustments. 

First off, for any single permutation, there are two ways to arrange "AB". The first is "AB" itself and the second is the reverse of that "BA". So we will multiply 3,628,800 by 2 to get 2*3,628,800 = 7,257,600

Using similar logic for "CD", we double 7,257,600 to get 2*7,257,600 = 14,515,200

The final answer is 14,515,200


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3 years ago
A health club charges non-members $5 per day to swim and $9 per day for an exercise class. Members pay a yearly fee of $300 plus
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The number of days d, he must use the gym to make the membership worthwhile is at least 21 days

Equation

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Members:

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= 21.42857142857142

Approximately,

21 days

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brainly.com/question/2972832

#SPJ1

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