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3 years ago

What is the correct way to label the entire line segment shown

Mathematics

1 answer:

Simora [160]3 years ago

7 0

Answer:Yea looks like you have it already C

Step-by-step explanation:

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The sum of the first n terms of a geometric series is 364? The sum of their reciprocals 364/243. If the first term is 1, find n

Afina-wow [57]

If the geometric series has first term $a$ and common ratio $r$ , then its $N$ -th partial sum is

$\displaystyle S_N = \sum_{n=1}^N ar^{n-1} = a + ar + ar^2 + \cdots + ar^{N-1}$

Multiply both sides by $r$ , then subtract $rS_N$ from $S_N$ to eliminate all the middle terms and solve for $S_N$ :

$rS_N = ar + ar^2 + ar^3 + \cdots + ar^N$

$\implies (1 - r) S_N = a - ar^N$

$\implies S_N = \dfrac{a(1-r^N)}{1-r}$

The $N$ -th partial sum for the series of reciprocal terms (denoted by $S'_N$ ) can be computed similarly:

$\displaystyle S'_N = \sum_{n=1}^N \frac1{ar^{N-1}} = \frac1a + \frac1{ar} + \frac1{ar^2} + \cdots + \frac1{ar^{N-1}}$

$\dfrac{S'_N}r = \dfrac1{ar} + \dfrac1{ar^2} + \dfrac1{ar^3} + \cdots + \dfrac1{ar^N}$

$\implies \left(1 - \dfrac1r\right) S'_N = \dfrac1a - \dfrac1{ar^N}$

$\implies S'_N = \dfrac{1 - \frac1{r^N}}{a\left(1 - \frac1r\right)} = \dfrac{r^N - 1}{a(r^N - r^{N-1})} = \dfrac{1 - r^N}{a r^{N-1} (1 - r)}$

We're given that $a=1$ , and the sum of the first $n$ terms of the series is

$S_n = \dfrac{1-r^n}{1-r} = 364$

and the sum of their reciprocals is

$S'_n = \dfrac{1 - r^n}{r^{n-1}(1 - r)} = \dfrac{364}{243}$

By substitution,

$\dfrac{1 - r^n}{r^{n-1}(1-r)} = \dfrac{364}{r^{n-1}} = \dfrac{364}{243} \implies r^{n-1} = 243$

Manipulating the $S_n$ equation gives

$\dfrac{1 - r^n}{1-r} = 364 \implies r (364 - r^{n-1}) = 363$

so that substituting again yields

$r (364 - 243) = 363 \implies 121r = 363 \implies \boxed{r=3}$

and it follows that

$r^{n-1} = 243 \implies 3^{n-1} = 3^5 \implies n-1 = 5 \implies \boxed{n=6}$

5 0

2 years ago

Solve f´(2) if f(x) = x^3 – 6x^2 + 9x + 3

Luba_88 [7]

Step-by-step explanation:

f(x) = x³ − 6x² + 9x + 3

Take the derivative and evaluate at x = 2.

f'(x) = 3x² − 12x + 9

f'(2) = -3

Check for local minimums or maximums by setting f'(x) equal to 0.

0 = 3x² − 12x + 9

0 = x² − 4x + 3

0 = (x − 1) (x − 3)

x = 1 or 3

Evaluate f(x) at the critical values, and at the end points.

f(0) = 3

f(1) = 7

f(3) = 3

f(5) = 23

f(x) has a minimum of 3 and a maximum of 23.

5 0

3 years ago

Two lines are guaranteed to be coplanar if they______. A. don't lie in the same plane B. lie in the same plane C. never meet D.

77julia77 [94]

B. lie in the same plane

5 0

3 years ago

11 x 14 = ?<br> What does 11 x 14 equal?

Darina [25.2K]

Answer:

154

Step-by-step explanation:

11 * 14

= 154

5 0

3 years ago

Read 2 more answers

Please please please can someone help me :)

disa [49]

Answer:

Step-by-step explanation:

now the equation looks like

3-4Cos(x)=y

hmmm i'm wondering if you can figure it out from here??? it's the same as the other two.. plug in that $\sqrt{2}$ /2 .. can you? :)

8 0

3 years ago

What is the correct way to label the entire line segment shown​

What is the correct way to label the entire line segment shown