summation

[lucid magnet]

It looks easier than it is I guess ,, hope for help!! Thx

Translation:

give the recursion as a sum using a summation factor and use this sum to calculate the value for x2. you do not have to break down the summation display any further

[native estuary]

x_n = -4x_{n-1} + 4 for every integer n >= 1, so

x_n = -4(-4x_{n-2} + 4) + 4 = (-4)^2 x_{n-2} + (-4)*4 + 4 =
= (-4)^2 (-4x_{n-3} + 4) + (-4)*4 + 4 =
= (-4)^3 x_{n-3} + (-4)^2*4 + (-4)*4 + 4 = ...

see the pattern

[spring loom]

Gillian Seed

[lucid magnet]

YESS

Got it

Finally

THANK YU

[tidal zealot]

Let’s find a constant sequence $(x_n)$ such that $x_n=-4x_{n-1}+2$. If $x_n=a$ for all $n\in\mathbb N$, then $a=-4a+2$ hence $a=\frac 25$.

[spring loom]

Gillian Seed

Let’s find a constant sequence $(x_n)$ such that $x_n=-4x_{n-1}+2$. If $x_n=a$ for all $n\in\mathbb N$, then $a=-4a+2$ hence $a=\frac 25$.
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[tidal zealot]

Anyway, let y_n=x_n-2/5

Where (x_n) is a sequence solving the proposed recurrence.

Then y_n=x_n-2/5=-4x_{n-1}+2-2/5=-4(y_{n-1}+2/5)+8/5=-4y_{n-1}

Hence y_n=(-4)^ny_0 for all n

y_0=x_0-2/5=-2/5

And x_n=2/5+y_n=2(1-(-4)^n)/5.

You’re welcome.
🥱

[balmy knot]

💀 bro they already got it lmao

[lucid magnet]

Ayo💀😭,,thx !