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
irga5000 [103]
3 years ago
12

Evaluate the sum of the following finite geometric series.

Mathematics
1 answer:
rjkz [21]3 years ago
5 0

Answer:

\large\boxed{\dfrac{156}{125}\approx1.2}

Step-by-step explanation:

<h3>Method 1:</h3>

\sum\limits_{n=1}^4\left(\dfrac{1}{5}\right)^{n-1}\\\\for\ n=1\\\\\left(\dfrac{1}{5}\right)^{1-1}=\left(\dfrac{1}{5}\right)^0=1\\\\for\ n=2\\\\\left(\dfrac{1}{5}\right)^{2-1}=\left(\dfrac{1}{5}\right)^1=\dfrac{1}{5}\\\\for\ n=3\\\\\left(\dfrac{1}{5}\right)^{3-1}=\left(\dfrac{1}{5}\right)^2=\dfrac{1}{25}\\\\for\ n=4\\\\\left(\dfrac{1}{5}\right)^{4-1}=\left(\dfrac{1}{5}\right)^3=\dfrac{1}{125}

\sum\limits_{n=1}^4\left(\dfrac{1}{5}\right)^{n-1}=1+\dfrac{1}{5}+\dfrac{1}{25}+\dfrac{1}{125}=\dfrac{125}{125}+\dfrac{25}{125}+\dfrac{5}{125}+\dfrac{1}{125}=\dfrac{156}{125}

<h3>Method 2:</h3>

\sum\limits_{n=1}^4\left(\dfrac{1}{5}\right)^{n-1}\to a_n=\left(\dfrac{1}{5}\right)^{n-1}\\\\\text{The formula of a sum of terms of a geometric series:}\\\\S_n=a_1\cdot\dfrac{1-r^n}{1-r}\\\\r-\text{common ratio}\to r=\dfrac{a_{n+1}}{a_n}\\\\a_{n+1}=\left(\dfrac{1}{5}\right)^{n+1-1}=\left(\dfrac{1}{5}\right)^n\\\\r=\dfrac{\left(\frac{1}{5}\right)^n}{\left(\frac{1}{5}\right)^{n-1}}\qquad\text{use}\ \dfrac{a^n}{a^m}=a^{n-m}\\\\r=\left(\dfrac{1}{5}\right)^{n-(n-1)}=\left(\dfrac{1}{5}\right)^{n-n+1}=\left(\dfrac{1}{5}\right)^1=\dfrac{1}{5}

a_1=\left(\dfrac{1}{5}\right)^{1-1}=\left(\dfrac{1}{5}\right)^0=1

\text{Substitute}\ a_1=1,\ n=4,\ r=\dfrac{1}{5}:\\\\S_4=1\cdot\dfrac{1-\left(\frac{1}{5}\right)^4}{1-\frac{1}{5}}=\dfrac{1-\frac{1}{625}}{\frac{4}{5}}=\dfrac{624}{625}\cdot\dfrac{5}{4}=\dfrac{156}{125}

You might be interested in
Explain how to write 3*3/8 as the product of a whole number and a unit fraction
Lady_Fox [76]
9/8 because you multiply 3 × 3 and leave the denominator the same
4 0
3 years ago
Vanessa deposited money into a bank account that earned 1.25% simple interest each year. After 1212 year, she had earned $5.00 i
Reil [10]
I=prt,,5.00=.0125*1212p,,p=5.00/.0125*1212=?? Use the calculator
3 0
3 years ago
68500 at 19% for 15 years what’s is the monthly payment
lubasha [3.4K]

Answer:

$1,599

Step-by-step explanation:

3 0
2 years ago
Please help me asap! sorry this si english =(
Ilia_Sergeevich [38]
1. Disappoint
2. Recall
3. Disgusted
4. Reunited
5. Prepare
7 0
2 years ago
A diagonal of a parallelogram forms 25° and 35° angles with the sides of a parallelogram. Find the angles of the parallelogram.
nata0808 [166]
Is this really still open?
8 0
3 years ago
Read 2 more answers
Other questions:
  • 1 1/4 as a improper fraction ​
    13·2 answers
  • Does 6(x+2)&gt;x-3 have a solution
    10·1 answer
  • HEllllppppppppoop!!!!!!!!!!!!!!!!!!!!!!!
    7·1 answer
  • In simplest radical form, what are the solutions to the quadratic eqaution 0=-3x^2-4x+5
    5·1 answer
  • PLEASE HELP
    7·1 answer
  • Find the missing number of each unit rate 48/3 = ?/1 56/8 = ?/1
    5·1 answer
  • Solve for the missing side to the nearest tenth.<br> 16<br> 30<br> X =
    12·1 answer
  • In a shop, a TV has a normal price of £500
    11·1 answer
  • 2. A cup of coffee costs $2.50. If the total cost
    15·1 answer
  • If F (x) =8+11 X-3X², find<br> a. What is F (5)? <br> b. What is F (x + b)? <br> c. What is F (-3)?
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!