1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
n200080 [17]
3 years ago
15

Find the sum of the geometric series 512+256+ . . .+4

Mathematics
1 answer:
mario62 [17]3 years ago
7 0

\bf 512~~,~~\stackrel{512\cdot \frac{1}{2}}{256}~~,~~...4

so, as you can see above, the common ratio r = 1/2, now, what term is +4 anyway?

\bf n^{th}\textit{ term of a geometric sequence}\\\\a_n=a_1\cdot r^{n-1}\qquad \begin{cases}n=n^{th}\ term\\a_1=\textit{first term's value}\\r=\textit{common ratio}\\----------\\r=\frac{1}{2}\\a_1=512\\a_n=+4\end{cases}

\bf 4=512\left( \cfrac{1}{2} \right)^{n-1}\implies \cfrac{4}{512}=\left( \cfrac{1}{2} \right)^{n-1}\\\\\\\cfrac{1}{128}=\left( \cfrac{1}{2} \right)^{n-1}\implies \cfrac{1}{2^7}=\left( \cfrac{1}{2} \right)^{n-1}\implies 2^{-7}=\left( 2^{-1}\right)^{n-1}\\\\\\(2^{-1})^7=(2^{-1})^{n-1}\implies 7=n-1\implies \boxed{8=n}

so is the 8th term, then, let's find the Sum of the first 8 terms.

\bf \qquad \qquad \textit{sum of a finite geometric sequence}\\\\S_n=\sum\limits_{i=1}^{n}\ a_1\cdot r^{i-1}\implies S_n=a_1\left( \cfrac{1-r^n}{1-r} \right)\quad \begin{cases}n=n^{th}\ term\\a_1=\textit{first term's value}\\r=\textit{common ratio}\\----------\\r=\frac{1}{2}\\a_1=512\\n=8\end{cases}

\bf S_8=512\left[ \cfrac{1-\left( \frac{1}{2} \right)^8}{1-\frac{1}{2}} \right]\implies S_8=512\left(\cfrac{1-\frac{1}{256}}{\frac{1}{2}}  \right)\implies S_8=512\left(\cfrac{\frac{255}{256}}{\frac{1}{2}}  \right)\\\\\\S_8=512\cdot \cfrac{255}{128}\implies S_8=1020

You might be interested in
NEED HELP 10 POINTS
leonid [27]

48.8 megabytes. When the download is 17.5% complete, have been downloaded 48.8 megabytes.

The key to solve this problem is using the rule of three.

If 279Mb -------------> 100%

      x      <-------------   17.5%

x = [(279 megabytes)(17.5)]/100

x = 48.825 megabytes

Round to the nearest tenth 48.8 megabytes.

6 0
3 years ago
Question is 15 points
Blababa [14]

sin(x+y)=sin(x)cos(y)-cos(x)sin(y)

also, remember pythagorean rule, sin^2(x)+cos^2(x)=1

given that sin(Θ)=4/5 and cos(x)=-5/13

find sin(x) and cos(Θ)


sin(x)

cos(x)=-5/13

using pythagorean identity

(sin(x))^2+(-5/13)^2=1

sin(x)=+/- 12/13

in the 2nd quadrant, sin is positve so sin(x)=12/13


cos(Θ)

sin(Θ)=4/5

using pythagrean identity

(4/5)^2+(cos(Θ))^2=1

cos(Θ)=+/-3/5

in 1st quadrant, cos is positive

cos(Θ)=3/5


so sin(Θ+x)=sin(Θ)cos(x)+cos(Θ)sin(x)

sin(Θ+x)=(4/5)(-5/13)+(3/5)(12/13)

sin(Θ+x)=16/65



answer is 1st option

7 0
3 years ago
The computer game Peter wants to buy will cost at least $50 and not more than $70. He earns $3 an hour running errands for his g
Lunna [17]

Answer:

The answer to your question is C

Just try it out

- have a nice day, be safe & healthy -

- peace out -

6 0
3 years ago
Read 2 more answers
Name three chords. <br><br><br>name five radii​
WITCHER [35]

Answer:

The three chords are AB , BD and EF

The five radii are AP, BP, EP, FP and CP

3 0
1 year ago
Read 2 more answers
Please help with my next question in math.
Vesnalui [34]
The answer is D. 2n + 1
8 0
3 years ago
Other questions:
  • What is 87 and 46 rounded to the nearest place value?
    7·1 answer
  • Which of the following is an equation for the line that passes through the point (0,0) and is perpendicular to the line shown ab
    11·2 answers
  • Use an addition sentence to help solve -30 -10
    11·2 answers
  • 3х + 5у = -3<br>x+2y = - 4/3​
    7·1 answer
  • The area of each triangular face of this right triangular prism is 8cm^2(squared).
    14·1 answer
  • How many pizzas were there at the start of the party?
    6·1 answer
  • What is the whole number for 9/10
    6·2 answers
  • A rectangular field has a length of x metres.
    14·1 answer
  • Chukwu, Abdul and kunle are to share $142, such that Abdul gets $11.00 less than Chukwu and $7.00 more than Kunle. How much is C
    10·1 answer
  • Hey! i’ll give brainliest help thanks
    9·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!