Mathematics, 16.12.2019 22:31 cheergirl21

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6. given the series: 4 + 12 + 36 +

find the 11th term of the series and find the sum of the first 11 terms.

7. a ball is dropped from a height of 40 feet and bounces back up to 90% of its previous height on each successive bounce. how far will the ball have traveled by the time it comes to a stop?

8. for each series below, decide whether it converges or diverges. find the sum, if it exists.

a) 80 + 20 + 5 + 5/4 +
b) 2/9 + 4/3 + 8 +

Answers: 2

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Answers

Answer from: jjdj5218

Q6. a₁₁ = 236 196; S₁₁ = 354 292

Q7. 354 292

Q8. a) Converges, S = 320/3; b) diverges

Step-by-step explanation:

6. Geometric sequence

The first three terms of your sequence are 4, 12, 36.

Each term differs from the previous one by a factor of 3, so it is a geometric sequence.

Each term has the form

aₙ = a₁rⁿ⁻¹

In your sequence, a₁ = 4 and r = 3.

Thus, the formula for the nth term is

aₙ = 4(3)ⁿ⁻¹

The 11th term is

a₁₁ = 4(3)¹¹⁻¹ = 4(3)¹⁰ = 4 × 59 049 = 236 196

The formula for the sum of the first n terms of a geometric series is

Sum = a₁[(1 - rⁿ)/(1 - r)]

For the sum over the first 11 terms,

Sum = 4[(1 - 3¹¹)/(1 - 3)

= 4(1 - 177 147)/(-2)

= -2(-177 146)

= 354 292

7. Bouncing ball

Mathematically, the ball never stops bouncing. The height of each bounce just gets infinitesimally small.

So , you have an infinite geometric series in which the first term is 40 ft and each successive term is 90 % of the previous term.

The general formula for the nth term is

aₙ = a₁rⁿ⁻¹

with a₁ = 40 ft and r = 0.90.

Since |r| <1, we have a convergent series, and the formula for the sum is

S = a₁/(1-r)

∴ S = 40/(1 - 0.90) = 40/0.10 = 400 ft

The bouncing ball will have travelled 400 ft.

8. Test for convergence

a) 80 + 20 + 5 + 5/4 + ...

r = a₂/a₁ = 20/80 = ¼

r < 1, so the series converges.

S = 80/( 1 - ¼) = 80/¾ = 80 × ⁴/₃ = 320/3

The sum of the series is 320/3.

b) 2/9 + 4/3 + 8 + ...

r = a₂/a₁ = (⁴/₃)/(²/₉) = ⁴/₃ × ⁹/₂ = 2 × 3 = 6.

r > 1, so the series diverges.

Answer from: jakeyywashere

6 236,196 and 354,294

Step-by-step explanation:

$\text{Find the 11th term:}\\a_1=4\qquad r=\dfrac{12}{4}=3\qquad n=11\\\\a_n=a_1\cdot r^{n-1}\\a_{11}=4\cdot (3)^{11-1}\\.\quad =4\cdot3^{10}\\.\quad =4\cdot 59,049\\.\quad =\large\boxed{236,196}\\\\\\\text{Find the sum:}\\S_n=\dfrac{a_1(1-r^n)}{1-r}\\\\\\S_{11}=\dfrac{4(1-3^{11})}{1-3}\\\\\\.\quad =\dfrac{4(1-177,147)}{-2}\\\\.\quad =-2(-177,146)\\\\.\quad =\large\boxed{354,294}$

7 400

Step-by-step explanation:

$\sum\limits^{n=1}_\infty 40(0.9)^{n-1}\\\\a_1=40\qquad r=0.9\\\\S_\infty =\dfrac{a_1}{1-r}\\\\\\.\quad =\dfrac{40}{1-0.9}\\\\\\.\quad =\dfrac{40}{0.1}\\\\\\.\quad =\large\boxed{400}$

8 a) converges

b) diverges

Step-by-step explanation:

A series converges if: -1 < r < 1

a) 80 + 20 + 5 + $\frac{5}{4}$

$a)\ r=\dfrac{a_2}{a_1}=\dfrac{20}{80}=\dfrac{1}{4}\implies \text{converges}\\\\\\b)\ r={a_2}\div{a_1}=\dfrac{4}{3}\div \dfrac{2}{9}=\dfrac{4}{3}\times \dfrac{9}{2}=2\times 3=6\implies \text{diverges}$

Answer from: Quest

The first option would be correct if it were one half and not one

Answer from: Quest

step-by-step explanation:

let

we know that

the y-intercept is the y-coordinate of the point when the value of x is equal to zero

the y-coordinate of the point a is the y-intercept

the answer part a) is

the value of y-intercept is

part b) find the slope m

we know that

the value of the slope is equal to

substitute the values

therefore

the answer part b) is

the value of the slope is equal to

the general answer is

the slope is

the value of y-intercept is

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Show work and explain with formulas.

6. given the series: 4 + 12 + 36 +

find...