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
eduard
2 years ago
9

Find the general term of {a_n}

Mathematics
1 answer:
Assoli18 [71]2 years ago
5 0

From the given recurrence, it follows that

a_{n+1} = 2a_n + 1 \\\\ a_{n+1} = 2(2a_{n-1} + 1) = 2^2a_{n-1} + 1 + 2 \\\\ a_{n+1} = 2^2(2a_{n-2}+1) + 1 + 2 = 2^3a_{n-2} + 1 + 2 + 2^2 \\\\ a_{n+1} = 2^3(2a_{n-3} + 1) + 1 + 2 + 2^2 = 2^4a_{n-3} + 1 + 2 + 2^2 + 2^3

and so on down to the first term,

a_{n+1} = 2^na_1 + \displaystyle \sum_{k=0}^{n-1}2^k

(Notice how the exponent on the 2 and the subscript of <em>a</em> in the first term add up to <em>n</em> + 1.)

Denote the remaining sum by <em>S</em> ; then

S = 1 + 2 + 2^2 + \cdots + 2^{n-1}

Multiply both sides by 2 :

2S = 2 + 2^2 + 2^3 + \cdots + 2^n

Subtract 2<em>S</em> from <em>S</em> to get

S - 2S = 1 - 2^n \implies S = 2^n - 1

So, we end up with

a_{n+1} = 4\cdot2^n + S \\\\ a_{n+1} = 2^2\cdot2^n + 2^n-1 \\\\ a_{n+1} = 2^{n+2} + 2^n - 1 \\\\\implies \boxed{a_n = 2^{n+1} + 2^{n-1} - 1}

You might be interested in
You invest $1300 in account that has an annual interest rate of 5% compounded annually. How much money will be in the account af
deff fn [24]
If you think about it and you use a calculator and do 1300×5% it will give you a number and divide it by 10 and there you should get you answer like that.
4 0
3 years ago
The nicotine content in a single cigarette of a particular brand has a distribution with mean 0.9 mg and standard deviation 0.1
Ad libitum [116K]

Answer:

The answer to the question is;

The probability that the resulting sample mean of nicotine content will be less than 0.89 is 0.1587 or 15.87 %.

Step-by-step explanation:

The mean of the distribution = 0.9 mg

The standard deviation of the sample = 0.1 mg

The size of the sample = 100

The mean of he sample = 0.89

The z score for sample mean is given by

Z =\frac{X-\mu}{\sigma/ \sqrt{n} } where

X = Mean of the sample

μ = Mean of the population

σ = Standard deviation of the population

Therefore Z = \frac{0.89-0.90}{0.1/\sqrt{100} } = -1

From the standard probabilities table we have the probability for  a z value of -1.0 = 0.1587

Therefore the probability that the resulting sample mean will be less than 0.89 = 0.1587 That is the probability that the mean is will be less than 0.89 is 15.87 % probability.

             

7 0
3 years ago
Find the term indicated in the expansion. (x - 3y)11; 8th term
Alja [10]
The term in the expansion:
T ( k+1) = n C k * A^(n-k) * B^k.
In this case: n = 11,  k + 1 = 8, so k = 7.
A = x, B = - 3 y
T 8 = 11 C 7 * x^(11-7) * ( - 3 y )^7 =
 =( 11 *10 * 9 * 8 * 7 * 6 * 5 ) / ( 7 * 6 * 5 * 4 * 3 * 2 * 1  )* x^4 * ( - 2,187 y^7 ) =
= 330 * ( - 2,187 ) x^4 y^7 = - 721,710 x^4 y^7
Answer: The 8th term in expansion is
 
8 0
3 years ago
Read 2 more answers
<img src="https://tex.z-dn.net/?f=%5Cfrac%7B5%7D%7B2%7D%20%3D%20%5Cfrac%7Bx%20-%204%7D%7B3%7D" id="TexFormula1" title="\frac{5}{
valentina_108 [34]

Step-by-step explanation:

\frac{5}{2}  =  \frac{x - 4}{3}

3 \times \frac{5}{2}  =  \frac{x - 4}{3}

\frac{3}{1}  \times  \frac{5}{2}  =  \frac{x - 4}{3}

cross multiplication

\frac{6 \times 5}{2}  =  \frac{x - 4}{3}

15 =  \frac{x - 4}{3}

15 \times 3 = x - 4

45 + 4 = x

49 = x

4 0
3 years ago
Read 2 more answers
Solve:7 +42-32-32 = 14.3-2a +7<br> A = -3<br> A= 0<br> A= 3<br> no solution
madreJ [45]

Answer:

a = 18.15

Step-by-step explanation:

7+42-32-32=14.3-2a+7

Combine like terms on the left side of the equation first. Add 7 and 42, then subtract 32 from the answer you get, then subtract 32 from that answer. This is going by order of PEMDAS.

Remember PEMDAS: (numbers 3 & 4 and numbers 5 & 6 are solved from left to right)

  1. Parentheses
  2. Exponents
  3. Multiplication
  4. Division
  5. Addition
  6. Subtraction

-15=14.3-2a+7

After combining like terms on the left side, you get -15. Now combine like terms on the right side of the equation by adding 14.3 and 7.

-15=21.3-2a

Get -2a alone by subtracting 21.3 from both sides.

-36.3=-2a

Divide both sides by -2.

a=18.15

In this equation, a should equal \boxed{18.15}.

5 0
3 years ago
Other questions:
  • Use the graph to find the slope of the line. Tell with the slope means in this context
    15·1 answer
  • Which country was Australia once a colony of?
    9·1 answer
  • HELP ASAP. MARK BRAINLIEST. 10-20 POINTS​
    14·1 answer
  • A grain trader buys the following amounts from three suppliers: 3,200 pounds, 7,200 pounds and 600 pounds. What is the total wei
    15·2 answers
  • Stephanie grew a total of 70 pounds of tomatoes to sell. She plans to sell the tomatoes for $3.50 per pound. The equation below
    8·1 answer
  • The graph of F(x), shown below, resembles the graph of G(x) x2, but it has
    9·1 answer
  • Find the constant of variation for the relation and use it to write an equation for the statement. Then solve the equation 7 If
    8·1 answer
  • Solve please I can’t figure out
    12·1 answer
  • Identify the radius and the center of a circle whose equation is (x – 5)² + y² = 81.
    8·1 answer
  • The image of the point (-7, 8) under a translation is (-11, 6). Find the coordinates
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!