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
dolphi86 [110]
2 years ago
5

The sum to infinity of a geometric series is 32.The sum of the first four terms is 30 and

Mathematics
1 answer:
gayaneshka [121]2 years ago
3 0

The terms in the series are

\displaystyle \sum_{n=1}^\infty ar^{n-1} = a + ar + ar^2 + ar^3 + \cdots

The series converges, so we |r|. All the terms in this particular series are known to be positive, so 0Consider the [tex]N-th partial sum of the series,

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

Multiply both sides by r :

r S_N = ar + ar^2 + ar^3 + \cdots + ar^N

Subtracting this from S_N eliminates all the r^n terms between n=1 and n=N-1, leaving us with

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

As N\to\infty, the r^N term will converge to zero, and the value of the infinite series is

\displaystyle \sum_{n=1}^\infty ar^{n-1} = \lim_{n\to\infty} S_N = S = \frac a{1-r}

Now, we're given

S = \dfrac a{1-r} = 32 ~~~~ \text{ and } ~~~~ S_4 = \dfrac{a(1-r^4)}{1-r} = 30

By substitution, we have

S_4 = 32(1-r^4) = 30 \implies 1-r^4 = \dfrac{15}{16} \implies r^4 = \dfrac1{16} \implies r = \dfrac12

so that

S = \dfrac a{1 - \frac12} = 32 \implies a = 16

Then the sum of the first 8 terms is

S_8 = \dfrac{a(1-r^8)}{1-r} = \dfrac{16\left(1 - \frac1{2^8}\right)}{1-\frac12} = \dfrac{255}8

and the difference between this and the infinite sum is

S - S_8 = 32 - \dfrac{255}8 = \dfrac{256-255}8 = \boxed{\dfrac18}

You might be interested in
Point C is between D and E and DE = 91 what is DC?
jok3333 [9.3K]

Answer:

DC=6

Step-by-step explanation:

all is shown and pictured

7 0
3 years ago
Multiply (5x2 + x − 4)(x + 2).
Trava [24]
(x + 2)(5x^2 + x - 4) =
x(5x^2 + x - 4) + 2(5x^2 + x - 4) =
5x^3 + x^2 - 4x + 10x^2 + 2x - 8 =
5x^3 + 11x^2 - 2x - 8 <===

6 0
3 years ago
Read 2 more answers
On a given blueprint, 1 inch = 30 feet. If the dimensions of a meeting room on the blueprint are 1.5 inches x 1.25 inches, what
melamori03 [73]
Sounds simple to me but may be misleading, I can't see a trick.
I think all you do is multiply each by 30.

So answer is 45ft X 37.5ft

Hope it's correct. I'd appreciate if you could let me know. Thks..
6 0
3 years ago
Solve.<br> x<br> __<br> 2 = 7<br> ----------------------------------------<br> PLEASE HELP!
galina1969 [7]
\frac{x}{2} = 7\\\\\frac{x}{2} \times 2 = 7\times2\\\\\boxed{x=14}
5 0
3 years ago
Can you guys please help!
yuradex [85]

Answer:

5 for the first and 4x for the second one also plz give heart

Step-by-step explanation:

i just know

4 0
3 years ago
Other questions:
  • FIRST PERSON TO ANSWER GETS BRANILYEST
    7·1 answer
  • How many plants x should she plant in each row so that her 25 plants end up in a square (i.e., x plants in each of x rows)?
    5·1 answer
  • Solve the system using substitution.<br> 5x+ 4y = 10<br> 4x - y = 8
    6·2 answers
  • If AC=18x-7,find the x. Round to the nearest hundredth if necessary <br>AB=7x+29 <br>BC=30​
    12·1 answer
  • Simplify the expression -1/2(-5/6 + 1/3)
    7·1 answer
  • What is 3x24 I think it is 72
    15·2 answers
  • ANYONE PLS HELP ASAP!!!
    11·2 answers
  • Pls help me frens......
    12·1 answer
  • HELP! EASY! Fill in the blanks!
    8·1 answer
  • Marcia has x number of homework problems. When she has done 10 of them, she is 1/4 finished. How many homework problems did she
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!