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
muminat
3 years ago
13

Mathematical induction, prove the following two statements are true

Mathematics
1 answer:
adelina 88 [10]3 years ago
3 0
Prove:
1+2\left(\frac12\right)+3\left(\frac12\right)^{2}+...+n\left(\frac12\right)^{n-1}=4-\dfrac{n+2}{2^{n-1}}
____________________________________________

Base Step: For n=1:
n\left(\frac12\right)^{n-1}=1\left(\frac12\right)^{0}=1
and
4-\dfrac{n+2}{2^{n-1}}=4-3=1
--------------------------------------------------------------------------

Induction Hypothesis: Assume true for n=k. Meaning:
1+2\left(\frac12\right)+3\left(\frac12\right)^{2}+...+k\left(\frac12\right)^{k-1}=4-\dfrac{k+2}{2^{k-1}}
assumed to be true.

--------------------------------------------------------------------------

Induction Step: For n=k+1:
1+2\left(\frac12\right)+3\left(\frac12\right)^{2}+...+k\left(\frac12\right)^{k-1}+(k+1)\left(\frac12\right)^{k}

by our Induction Hypothesis, we can replace every term in this summation (except the last term) with the right hand side of our assumption.
=4-\dfrac{k+2}{2^{k-1}}+(k+1)\left(\frac12\right)^{k}

From here, think about what you are trying to end up with.
For n=k+1, we WANT the formula to look like this:
1+2\left(\frac12\right)+...+k\left(\frac12\right)^{k-1}+(k+1)\left(\frac12\right)^{k}=4-\dfrac{(k+1)+2}{2^{(k+1)-1}}

That thing on the right hand side is what we're trying to end up with. So we need to do some clever Algebra.

Combine the (k+1) and 1/2, put the 2 in the bottom,
=4-\dfrac{k+2}{2^{k-1}}+\dfrac{(k+1)}{2^{k}}

We want to end up with a 2^k as our final denominator, so our middle term is missing a power of 2. Let's multiply top and bottom by 2,
=4+\dfrac{-2(k+2)}{2^{k}}+\dfrac{(k+1)}{2^{k}}

Distribute the -2 and combine the fractions together,
=4+\dfrac{-2k-4+(k+1)}{2^{k}}

Combine like-terms,
=4+\dfrac{-k-3}{2^{k}}

pull the negative back out,
=4-\dfrac{k+3}{2^{k}}

And ta-da! We've done it!
We can break apart the +3 into +1 and +2,
and the +0 in the bottom can be written as -1 and +1,
=4-\dfrac{(k+1)+2}{2^{(k-1)+1}}
You might be interested in
In a class of c children there are 16 boys what fraction of the class are boys​
lakkis [162]

Answer:

c/16

Step-by-step explanation:

7 0
3 years ago
A bank loans ​$2100 to a business at 7​% for 3 months. How much simple interest will the bank​ earn?
Natali [406]

Answer:

  $36.75

Step-by-step explanation:

The amount of interest payable is given by ...

  I = Prt = $2100·0.07·(3/12) . . . . . . . . t is in years

  I = $36.75

The bank earned $36.75 on the loan.

7 0
3 years ago
The ratio of Ed's toy cars to Pete's toy cars was initially 5:2. After Ed gave 30 toy cars to Pete, they each had an equal numbe
Slav-nsk [51]

Answer:

140 toy cars

Step-by-step explanation:

The ratio of Ed's toy car to Pete's toy car is initially given as 5:2

Ed gave Pete a total number of 30 cars

Let x represent the greatest common factor that exists between both number

Number of Ed's car is represented as 5x

Number of Pete car is represented as 2x

Since they each have an equal number of cars which is 30 then we can solve for x as follows

5x-30=2x+30

Collect the like terms

5x-2x= 30+30

3x= 60

Divide both sides by the coefficient of x which is 3

3x/3=60/3

x=20

Ed's car is 5x, we substitute 20 for x

5(20)

= 100 cars

Pete car is 2x,we substitute 20 for x

2(20)

= 40 cars

Therefore, the total number of cars can be calculated as follows

= 100+40

= 140 toy cars

Hence they have 140 toy cars altogether

5 0
3 years ago
Read 2 more answers
Jamey took a picture to use as a logo for his company. The picture is 24 in. wide  ×  36 in. high. The logo is 6 in. wide. 
ikadub [295]
I believe it would be 9 inches in height.
(1/4)*36=9
hope this helps!
6 0
3 years ago
Worth all my points plus brainliest!!! Due in 4 hrs 1-11
SVETLANKA909090 [29]

Answer:

1. 15

2. -8/9

3. 5

4. 7

5. a = 20

6. n = 100

7. 3.6 x 10^1

8. 3

9. 4

10. 8.37 x 10^5

11. 6.67 x 10^3

8 0
3 years ago
Other questions:
  • In the diagram of circle A, what is the measure of ∠XYZ?<br><br> 35°<br> 70°<br> 75°<br> 140°
    8·2 answers
  • Please help this is due really soon
    15·2 answers
  • There are 180 students in the gym for an assembly. if 5% of them are girls, how many girls are there
    11·2 answers
  • certain forest covers an area of 2600 km2 . Suppose that each year this area decreases by 6% . What will the area be after 13 ye
    11·1 answer
  • Nolan calculates his profit selling cakes by using the function p(c) = 7.5c -35, where c represents the number of cakes sold Whi
    5·1 answer
  • Four balls of wool make 8 knitted caps how much many balls of wool will Malcom
    9·1 answer
  • What is the axis of symmetry of the function f(x)= -(x+9)(x-21)
    14·1 answer
  • Solve: [4x + 2] + 6 =0
    11·2 answers
  • What is 3.5 divided by 7
    14·2 answers
  • Using two or more complete sentences, describe how to find the general rule for the geometric sequence with a4=3 and a6=3/16. in
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!