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
marin [14]
2 years ago
12

Solve the following recurrence relation:

%200" id="TexFormula1" title="A_{n}=a_{n-1}+n; a_{1} = 0" alt="A_{n}=a_{n-1}+n; a_{1} = 0" align="absmiddle" class="latex-formula">
Mathematics
1 answer:
-Dominant- [34]2 years ago
4 0

By iteratively substituting, we have

a_n = a_{n-1} + n

a_{n-1} = a_{n-2} + (n - 1) \implies a_n = a_{n-2} + n + (n - 1)

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

and the pattern continues down to the first term a_1=0,

a_n = a_{n - (n - 1)} + n + (n - 1) + (n - 2) + \cdots + (n - (n - 2))

\implies a_n = a_1 + \displaystyle \sum_{k=0}^{n-2} (n - k)

\implies a_n = \displaystyle n \sum_{k=0}^{n-2} 1 - \sum_{k=0}^{n-2} k

Recall the formulas

\displaystyle \sum_{n=1}^N 1 = N

\displaystyle \sum_{n=1}^N n = \frac{N(N+1)}2

It follows that

a_n = n (n - 2) - \dfrac{(n-2)(n-1)}2

\implies a_n = \dfrac12 n^2 + \dfrac12 n - 1

\implies \boxed{a_n = \dfrac{(n+2)(n-1)}2}

You might be interested in
the cost of an air lift ride in Gatlinburg is 5$ for the first 30 minutes and $6 for each additional hour. what is the cost of a
UkoKoshka [18]

Answer:

88

Step-by-step explanation:

5+(6x8)=88

6 0
3 years ago
PLEASE HELP ME! GIVING POINTS!
creativ13 [48]
11. 4 + 10 = 14 
25 - 14 = 11
3 0
2 years ago
Read 2 more answers
Patti's Pet Palace has a 90-gallon fish tank. When it needs to be cleaned, it can be drained at a rate of 2 gallons per minute.
dangina [55]
Y = 90 - 2x

This should be the correct answer
6 0
3 years ago
Read 2 more answers
The girls soccer team won 10 games and the lost 2. And the boys soccer team won 12 games and lost 3. What is the ratio of the bo
Eduardwww [97]
The ratio of the boys is 4/1
8 0
3 years ago
Read 2 more answers
Based on the given segment lengths, determine whether the segments can form a triangle.
murzikaleks [220]

For 3 sides to form a triangle two sides added together need to be greater then the 3rd side. You need to check all 3 combinations:

1. 11.5 + 10.5 = 22, 22> 6.1

11.5 + 6.1 = 17.6 17.6 > 10.5

10.5 + 6.1 = 16.6, 16.6 > 11.5

This one can make a triangle.

2. 9.5 + 3.5 > 4.5

9.5 + 4.5 > 3.5

4.5 + 3.5 < 9.5

Cannot make a triangle.

3. 8 + 6.7 > 1

6.7 + 1 < 8

Cannot make a triangle.

4. Can make a triangle

5. Can

6. Can

4 0
3 years ago
Other questions:
  • Can anyone help with number 54?
    14·1 answer
  • What is the slope of a line that is perpendicular to the line y = 8x + 5?
    8·2 answers
  • 15 POINTS!
    9·1 answer
  • How can you tell from the structure of the equation that it is quadratic?
    14·1 answer
  • Can the product of a whole number and a decimal number less than 1 ever be greater than a whole number
    12·1 answer
  • The triangular prism below has a base area of 9 cm and a height of 23 cm. What is the volume of the prism?A. 207 cm B. 103.5 cm
    14·1 answer
  • On a farm, 34 of each acre is planted with blue spruce trees. If there are 16 12 acres of trees on the farm, how many acres are
    7·1 answer
  • Hey, can you help me please and thank you
    11·2 answers
  • A landscaping business is purchasing plants from a nursery to finish a project. It will cost $14 each to buy each tree and $5 ea
    14·1 answer
  • HELPPP ASAPPPPPPP !!!!!!
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!