Question

Solve the recurrence relation , an–7an-1+10an-2= 3n, given that a0= 0 and a1= 1

Answer 1

Answer:

Solve the recurrence relation a(n)=7a(n−1)−10a(n−2)

Not really sure where to start. I see that I can generalize it a bit to be

a(n)=7(7(a(n−1)−10a(n−2)−10(7(a(n−3))−10(a(n−4)))

Answer 2

$\displaystyle \bf a_n = \frac{1}{3} \bigg( {5}^{n} - {2}^{n}\bigg) \: \: \: for \: al l\: \: n \geqslant 0$

Which is the required solution of the recurrence relation

Given :

The recurrence relation $\sf{a_n - 7a_{n - 1} + 10a_{n - 2} = 0}$ with $\sf{a_0 = 0 \: \: \: and \: \:a_1 = 1 }$

To find :

The solution of the recurrence relation

Solution :

Step 1 of 2 :

Write down the given recurrence relation

The recurrence relation

$\sf{a_n - 7a_{n - 1} + 10a_{n - 2} = 0}$ with $\sf{a_0 = 0 \: \: \: and \: \:a_1 = 1 }$

Step 2 of 2 :

Solve the recurrence relation

Here the given recurrence relation is

$\sf{a_n - 7a_{n - 1} + 10a_{n - 2} = 0}$

This is a second order homogeneous difference equation with constant coefficients

The corresponding auxiliary equation is

$\sf{ {x}^{2} - 7x + 10 =0 }$

$\implies \sf{ \sf{ (x - 2)(x - 5) = 0 }}$

$\implies \sf{ \sf{x = 2 \:, \: 5}}$

The roots of the auxiliary equation are real and distinct

∴ There are two constants b and c such that

$\sf{a_n = b .\: {2}^{n} +c. \: {5}^{n} \: \: \: \: \: \: \: for \: all \: n \geqslant 0 }$

Now the initial conditions are

$\sf{a_0 = 0 \: \: \: and \: \:a_1 = 6 }$

Now

$\sf{a_0 = 0 \: \: gives \: \: b + c = 0 }$

$\sf{a_1 = 1\: \: \: gives \: \: \:2b + 5c = 1 }$

Above gives

$\sf{ - 2c + 5c = 1 }$

$\implies \sf{ 3c = 1 }$

$\displaystyle \sf{ \implies }c = \frac{1}{3}$

$\displaystyle \sf{ \implies }b = - c = - \frac{1}{3}$

Now

$\sf{a_n = b .\: {2}^{n} +c. \: {5}^{n} \: \: \: \: \: \: gives\: }$

$\displaystyle \sf a_n = - \frac{1}{3} .\: {2}^{r} + \frac{1}{3} . \: {5}^{r}$

$\displaystyle \sf{ \implies }a_n = \frac{1}{3} \bigg( {5}^{n} - {2}^{n}\bigg)$

Hence the required solution is given by

$\displaystyle \sf a_n = \frac{1}{3} \bigg( {5}^{n} - {2}^{n}\bigg) \: \: \: for \: al l\: \: n \geqslant 0$

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