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3 years ago

Which formula can be used to find the nth term of a geometric sequence where the fifth term isn1/16 and the common ratio is 1/4?

Mathematics

1 answer:

kati45 [8]3 years ago

7 0

1/16=a(1/4)^(5-1)

1/16=a(1/4)^4

1/16=a/256

256/16=a

16=a

So the initial term is 16 so our formula is:

a(n)=16(1/4)^(n-1)

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There are 25 questions on your math test. You have completed 5 questions so far. What fraction of the total questions do you sti

maksim [4K]

Answer:

$\frac{4}{5}$

Step-by-step explanation:

If you have completed $5$ questions, you still have $25-5=20$ more questions to answer. We need to find what fraction $20$ is of $25$ , if that makes sense. A fraction is simply a part over a whole. In this case, the part is $20$ and the whole is $25$ . Therefore, the fraction we are looking for is $\frac{20}{25}$ , which simplifies to $\frac{4}{5}$ . Hope this helps!

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2 years ago

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The binomial (2x+6) is a factor of a quadratic expression 10x^2+18x-36 what is the other factor of this expression?

madreJ [45]

Answer:

The other factor of the polynomial is (5x -6)

Step-by-step explanation:

Here, we want to find the other factor of this binomial expression

To find the other factor, we simply proceed to divide the polynomial by the binomial

That would be;

(10x^2 + 18x - 36)/(2x + 6)

= 5x - 6

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2 years ago

Compare the numbers using <, >, or =. 0.78 ___ 0.708 < > =

taurus [48]

Compare the numbers using <, >, or =. 0.78 ___ 0.708 < > =

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3 years ago

Ramon is a waiter. He waits on a table of 4 whose bill comes to $99.69. If Ramon receives a 20% tip, approximately how much will

vlabodo [156]

99.69/100=.9969
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3 years ago

How to know if a function is periodic without graphing it ?

zhenek [66]

A function $f(t)$ is periodic if there is some constant $k$ such that $f(t+k)=f(k)$ for all $t$ in the domain of $f(t)$ . Then $k$ is the "period" of $f(t)$ .

Example:

If $f(x)=\sin x$ , then we have $\sin(x+2\pi)=\sin x\cos2\pi+\cos x\sin2\pi=\sin x$ , and so $\sin x$ is periodic with period $2\pi$ .

It gets a bit more complicated for a function like yours. We're looking for $k$ such that

$\pi\sin\left(\dfrac\pi2(t+k)\right)+1.8\cos\left(\dfrac{7\pi}5(t+k)\right)=\pi\sin\dfrac{\pi t}2+1.8\cos\dfrac{7\pi t}5$

Expanding on the left, you have

$\pi\sin\dfrac{\pi t}2\cos\dfrac{k\pi}2+\pi\cos\dfrac{\pi t}2\sin\dfrac{k\pi}2$

and

$1.8\cos\dfrac{7\pi t}5\cos\dfrac{7k\pi}5-1.8\sin\dfrac{7\pi t}5\sin\dfrac{7k\pi}5$

It follows that the following must be satisfied:

$\begin{cases}\cos\dfrac{k\pi}2=1\\\\\sin\dfrac{k\pi}2=0\\\\\cos\dfrac{7k\pi}5=1\\\\\sin\dfrac{7k\pi}5=0\end{cases}$

The first two equations are satisfied whenever $k\in\{0,\pm4,\pm8,\ldots\}$ , or more generally, when $k=4n$ and $n\in\mathbb Z$ (i.e. any multiple of 4).

The second two are satisfied whenever $k\in\left\{0,\pm\dfrac{10}7,\pm\dfrac{20}7,\ldots\right\}$ , and more generally when $k=\dfrac{10n}7$ with $n\in\mathbb Z$ (any multiple of 10/7).

It then follows that all four equations will be satisfied whenever the two sets above intersect. This happens when $k$ is any common multiple of 4 and 10/7. The least positive one would be 20, which means the period for your function is 20.

Let's verify:

$\sin\left(\dfrac\pi2(t+20)\right)=\sin\dfrac{\pi t}2\underbrace{\cos10\pi}_1+\cos\dfrac{\pi t}2\underbrace{\sin10\pi}_0=\sin\dfrac{\pi t}2$

$\cos\left(\dfrac{7\pi}5(t+20)\right)=\cos\dfrac{7\pi t}5\underbrace{\cos28\pi}_1-\sin\dfrac{7\pi t}5\underbrace{\sin28\pi}_0=\cos\dfrac{7\pi t}5$

More generally, it can be shown that

$f(t)=\displaystyle\sum_{i=1}^n(a_i\sin(b_it)+c_i\cos(d_it))$

is periodic with period $\mbox{lcm}(b_1,\ldots,b_n,d_1,\ldots,d_n)$ .

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2 years ago