### Solving a Recurrence Relation**Problem Statement:**Solve the recurrence relation $a_{n+1} = 7a_n - 10a_{n-1}$, for $ n \geq 2 $, given the initial conditions $a_1 = 10$ and $a_2 = 29$.**Solution Approach:**To solve this recurrence relation, we can use characteristic equations:1. **Form the Characteristic Equation:** \[ r^2 = 7r - 10 \] Rearrange to get: \[ r^2 - 7r + 10 = 0 \]2. **Solve the Characteristic Equation:** Factorize the quadratic equation: \[ (r - 5)(r - 2) = 0 \] Thus, the roots are: \[ r_1 = 5, \quad r_2 = 2 \]3. **General Solution:** Using the roots, the general solution for the recurrence relation is: \[ a_n = A \cdot 5^n + B \cdot 2^n \]4. **Find Constants $A$ and $B$:** Use the initial conditions to form a system of equations: \[ \begin{cases} 10 = A \cdot 5^1 + B \cdot 2^1 \\ 29 = A \cdot 5^2 + B \cdot 2^2 \\ \end{cases} \] Simplify them: \[ \begin{cases} 10 = 5A + 2B \\ 29 = 25A + 4B \\ \end{cases} \]5. **Solve the System of Equations:** Multiply the first equation by 2: \[ 20 = 10A + 4B \] Subtract this from the second equation to eliminate $B$: \[ 29 - 20 = 25A - 10A \] This simplifies to: \[ 9 = 15A \quad \Rightarrow \quad A = \

Name: ### Solving a Recurrence Relation**Problem Statement:**Solve the recu
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: ### Solving a Recurrence Relation**Problem Statement:**Solve the recurrence relation $a_{n+1} = 7a_n - 10a_{n-1}$, for $ n \geq 2 $, given the initial conditions $a_1 = 10$ and $a_2 = 29$.**Solution Approach:**To solve this recurrence relati

Answered: Image /qna-images/answer/c05d82e7-8a59-4161-87fa-b1a7439be1fa.jpg

Answered: Solve the recurrence relation an+1 =…

Math

Advanced Math

Solve the recurrence relation an+1 = 7an – 1Oan-1,n > 2, given a1 = 10, a2 = 29.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Solve the recurrence relation an = an-1 +6an-2 for all integers n ≥ 2, with a = 3, a₁ = 3, a₁ = 9.

Q: Given the recurrence relation an =-2an-1 +15an-2 with the initial conditions ao=1 and a1=7. (a)…

A: Given the recurrence relation an=-2an-1+15an-2 with the initial conditions a0=1 and a1=7 (a) We…

Q: Find the solution to the following recurrence: an = 6an-18an-2+ 9n for n ≥ 2 with initial conditions…

Q: Consider the recurrence relation an = 5an-1 + 14an-2 with first two terms a0 = 10 and a1 = 18. Solve…

A: Introduction: Recursion is used in a recurrence relation to connecting phrases in a sequence of…

Q: Solve the linear recurrence relation a, = 8a,-16a,-2,ɑ, = 1,a, =1 %3D

A: To find the solution of the linear recurrence relation of given second degree first we need to find…

Q: or, Recurrence Formulae : ¨ (2n + 1) xP₂ = (n + 1) Pn+j+nP,-1 (n+1) Pn+1 = (2n + 1) xP₁ − nP,. -

A: ⁰0⁰

Q: Consider the recurrence relation an = 3an-1 + 10an-2 with first two terms ao = 5 and = 6. a₁ = a.…

A: This question related to discrete mathematics.

Q: Use the characteristic root technique to solve the following recurrence relations ( find a…

Q: Solve the linear recurrence relation a, = 5a- 6a , a, =7, a, =16, for all n2 2. %3D

Q: Solve the recurrence relation for n greater than or equal to 3, and e0=e1=1 and e2=2.…

Q: Solve the recurrence relation using the generating function technique. b0=1/2 bn=3bn-1 - 1/2…

A: To solve : recurrence relation using the generating function technique bn=3bn-1-12 ,…

Q: Solve the following recurrence relation: x(n) = x(n-1) + n for n > 0, and x(0) = 0.

Q: 3. Let, a1 = 3, a2 = 4 and for n > 3, an = 2an-1+an-2 – 5, express an in terms of n.

Q: Find the solution to the recurrence relation an = 3an-1 + 184,-2 with initial terms ao = 2 and a₁ =…

Q: an = 2n + an-1, Co=2

A: Observing the pattern by putting the values of n = 1, 2, 3............n.

Q: Solve the recurrence relation an an-1+8an-2-12an-3, n ≥ 3, subject to the initial conditions ao 0,…

Q: 1. Solve: An = An -1+ n; Ao = 0

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: Solve the with ao = 7. an recurrence relation an = = An-1 + 2n Preview

Q: Find a solution to the following recurrence: -11an-1 an = an - 28an-2-5(-2)" for n ≥ 2 with initial…

Q: Solve the recurrence relation USING THE GENERATING FUNCTION for n greater than or equal to 3, and…

Q: an 64a,-2 for n > 2 with the initial condition ao :-16, a1 -96. an I| ||

Q: 17) Solve the recurrence relation: 1 b, = 4 b, = 2b,-1 - bp2 %3D n-2

A: Given, the recurrence relation with . This is a linear homogenous recurrence relation, Thus the…

Q: Solve the recurrence equation an = 4an-1 + 21an-2 + 2 - 3 * 4n where a1 = 5 and a0 = 2

Q: Solve the recurrence relation an with initial terms ao ат -1+12аn -2 - 7. %3D 7 and a1 = - An =

Q: Solve the recurrence equation an = 4an-1 + 21an-2 + 2 - 3 * 4n where a1 = 5 and a0 = 2

Q: Let C be the curve of intersection of the surfaces y = V2 x2 and z 2/2 ху. (a) Compute the curvature…

A: Given C be the curve of intersection of the surfaces y=2x2 and z=223xy

Q: Solve the recurrence relation: An = 8an/2 + 6n with a2 = 12. Assume n has a power of 2.

A: Given recurrence relation can be written as: An=8An2+6n This relation is of the form: An=aAnb+fn…

Q: Consider the recurrence relation an = an-1 +2an−2 +2n−9 for n ≥ 2, a0 = 2, a1 = 1. Show that an = −n…

Q: 8 a) Solve the recurrence relation an an-1 + an-2, n 22, ao = 1, a = 2 %3D

A: As per our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: 1. Write the recurrence an+2 = 4a,+1+21a, using the advancement operator. Then solve for a, using…

A: First we recurrence using advancement operator then we find an

Q: Solve the recurrence relation an 10an-1, a0 = 3. Choose the most appropriate option. O r= 10, a = 3…

Q: Find the solution to the following Ihcc recurrence: 6an-1 + 7an-2 for n > 3 with initial conditions…

Q: an 34an-1 – 289an-2 for n > 3 with initial conditions aj = 68, az = -1734. an

Q: Find the unique, closed form solution to this recurrence relation given the initial condition: an =…

Q: Solve the recurrence relation ba = 8bn-1 - 16b,-2 +4 for n2 2 with initial values bo = 1, b1 = 0. 山C

Concept explainers

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

Topic Video

Question

### Solving a Recurrence Relation

**Problem Statement:**
Solve the recurrence relation $a_{n+1} = 7a_n - 10a_{n-1}$, for $ n \geq 2 $, given the initial conditions $a_1 = 10$ and $a_2 = 29$.

**Solution Approach:**
To solve this recurrence relation, we can use characteristic equations:

1. **Form the Characteristic Equation:**
\[
r^2 = 7r - 10
\]
Rearrange to get:
\[
r^2 - 7r + 10 = 0
\]

2. **Solve the Characteristic Equation:**
Factorize the quadratic equation:
\[
(r - 5)(r - 2) = 0
\]
Thus, the roots are:
\[
r_1 = 5, \quad r_2 = 2
\]

3. **General Solution:**
Using the roots, the general solution for the recurrence relation is:
\[
a_n = A \cdot 5^n + B \cdot 2^n
\]

4. **Find Constants $A$ and $B$:**
Use the initial conditions to form a system of equations:
\[
\begin{cases}
10 = A \cdot 5^1 + B \cdot 2^1 \\
29 = A \cdot 5^2 + B \cdot 2^2 \\
\end{cases}
\]
Simplify them:
\[
\begin{cases}
10 = 5A + 2B \\
29 = 25A + 4B \\
\end{cases}
\]

5. **Solve the System of Equations:**
Multiply the first equation by 2:
\[
20 = 10A + 4B
\]
Subtract this from the second equation to eliminate $B$:
\[
29 - 20 = 25A - 10A
\]
This simplifies to:
\[
9 = 15A \quad \Rightarrow \quad A = \

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Recall:

VIEW

Solution:

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Propositional Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Solve the recurrence relation, given: • ao = 3 • a₁ = 4 . • an = 5an-1-6an-2 Remember: • The theorem is an = c₁an-1 + c₂an-2. -b± √b² - 4ac 2a • The quadratic formula is O a O b Oc Od an = 5(4)" -2(3)" an = 5(2)"-2(3)" an = 5(2)"+2(3)" an = 5(3)" -2(3)"
Solve the recurrence relation: an = 6an-1 -9an-2 %3D 00 =D4 0; = 15
Solve the recurrence relation, bn - 5b = 0,n >= 1,b0 = 1
Q2. Solve the recurrence relation an = ao = 1 and a, 7an-1 - 10 an-2 with initial conditions. = 2.
Solve the following recurrence relations. (b) an = 2an-1 — an-2 with að = a₁ = = 2
Solve this recurrence relation together with the initial conditions given an+2=-4an+1+5an, for n≥0, a0=2, a1=8
Solve the recurrence relation an = −3an−1 given that a0 = 1.
Solve the recurrence relation an = 7an-1 -12an-2 for all integers n ≥ 2, a = 1 and a₁ = 6.