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3 years ago

If the first 4 terms of a geometric sequence are {7, 21, 63, 189), then the formula for the nth term in the sequence is?

Mathematics

1 answer:

34kurt3 years ago

7 0

Answer: 7 × ( 3^n - 1 )

Step-by-step explanation:

This is a geometric progression which has its nth term to be

Tn = ar^n- 1. Now from the question given, the sequence are 7, 21, 63, 189. This is an example of a finite sequence, To find the common ratio, you divide the second term by the first term. So

r = 21/7

= 3. , a = 7

Now to find the nth term put the values in the formula above.

Tn =ar^n-1

= 7(3)^n - 1)

= 7 × ( 3^n - 1 ).

Note, it is not 21^n - 1.

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Given LaTeX: f\left(x\right)=x^{^3}-3x+4f ( x ) = x 3 − 3 x + 4, determine the intervals where the function is increasing and wh

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Increasing: $x$ and $x>1$ .

Decreasing: $-1$

Step-by-step explanation:

We have been given a function $f(x)=x^3-3x+4$ . We are asked to determine the intervals, where the function is increasing and where it is decreasing.

First of all, we will find critical points of our given function by equating derivative of our given function to 0.

Let us find derivative of our given function.

$f'(x)=\frac{d}{dx}(x^3)-\frac{d}{dx}(3x)+\frac{d}{dx}(4)$

$f'(x)=3x^{3-1}-3+0$

$f'(x)=3x^{2}-3$

Let us equate derivative with 0 as find critical points as:

$0=3x^{2}-3$

$3x^{2}=3$

Divide both sides by 3:

$x^{2}=1$

Now we will take square-root of both sides as:

$\sqrt{x^{2}}=\pm\sqrt{1}$

$x=\pm 1$

$x=-1,1$

We know that these critical points will divide number line into three intervals. One from negative infinity to -1, 2nd -1 to 1 and 3rd 1 to positive infinity.

Now we will check one number from each interval. If derivative of the point is greater than 0, then function is increasing, if derivative of the point is less than 0, then function is decreasing.

We will check -2 from our 1st interval.

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