Sequence

[pine falcon]

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[outer stream]

i’m stuck on this

for for 30mns now

[hollow knot]

isn't this just the fibonacci sequence

alr lemme try

[outer stream]

idk what that is

[hollow knot]

oh oof

[outer stream]

i just wanna get through 11th grade

[hollow knot]

lol

[craggy basin]

oh then its prolly AP base

[hollow knot]

alr I'll help
just lemme study the question

[hollow knot]

no it isn't an AP

[craggy basin]

oh ok

[outer stream]

thank you

[hollow knot]

I think you'll have to use an inductive approach here

I'll try that

[sick crane]

@outer stream has given 1 rep to @hollow knot

[hollow knot]

you also try it

[outer stream]

wdym?

[hollow knot]

hmmm

wait a min

how would you prove $\sum_{n=1}^{k}n = \frac{k(k+1)}{2}$

[shrewd moat]

Inverse Cupid

[hollow knot]

for all natural k?

[outer stream]

imma be honest man i dont know

[hollow knot]

oh...

ok then

[outer stream]

i haven’t learned abt series yet

its the lesson after sequences

[hollow knot]

I'll post the solution for your question

and then walk you through step by step

[outer stream]

Okay yes

[hollow knot]

Given: $a_{n+2} = a_{n+1} + a_n, a_0=0, a_1=1$

TP $a_{n+2} = 1 + \sum^{n}_{k=0}a_k$ :

Let's first check for a few values of n:

if $n = 0$:

$a_2 = a_0 + 1$

$a_1 = a_0 + 1$

$1 = 1$

now for $n=1$:

$a_3 = a_0 + a_1 + 1$

$a_2 + a_1 = a_0 + a_1 + 1$

$1 + 1 = 0 + 1 + 1$

Now, let's assume the following is true for $n=k$:

So, $a_{k+2} = 1 + \sum^{k}_{j=0}a_j$

Now, let's prove it for $n=k+1$:

$a_{k+3} = 1 + \sum^{k+1}_{j=0}a_j$

$a_{k+2} + a_{k+1} = 1+ \sum^{k+1}_{j=0}a_j$

$a_{k+2} = 1+ \sum^{k}_{j=0}a_j$

The above is what we assumed to be true

As we derived the above from trying to prove for $n=k+1$

By the hypothesis of induction

hence proved.

[shrewd moat]

Inverse Cupid

[hollow knot]

induction works like dominoes

we proved it's true for 1, 2 and k+1 for any value of k
so if k=2, then it's also true for 3
now, if k=3, it's also true for 4
and so on till positive infinity

[outer stream]

wait lemme try to understand

[hollow knot]

alr

[outer stream]

why is a2 = a0 + 1

where did the 1 come from

[hollow knot]

it's in the question

a(n+2) = 1+a0+a1+a2+...+a(n)

[outer stream]

so does that mean
a2 = 1 + an

[hollow knot]

and n is 0

so

a2 = 1 + a0

ryt?

[outer stream]

but since
a(n+2) = 1 + a0 + a1 + … + an
for n = 0
isn’t it supposed to be
a2 = 1 + a0 + a1 + …. + a0 ?

[hollow knot]

lol

no it isn't like that

[outer stream]

😭😭

[hollow knot]

I thought that's where you went confused though

the dot dot dot just means upto

[outer stream]

OH MY GOD

I just understood

[hollow knot]

upto a(n)

[outer stream]

i’m so dum

[hollow knot]

nah you're not

[outer stream]

wait gonna read it over again

[hollow knot]

alr

[outer stream]

is this really a proof?

it’s kinda like a checking

Idk

[hollow knot]

yeah it's a way of proving

for some reason

it's called induction in mathematics

but it just works like dominoes

[outer stream]

well i haven’t learned about series yet so the bottom half is kinda confusing for me

[hollow knot]

oh the sigma part huh

[outer stream]

is there a different way we can write it?

[hollow knot]

that's just compact notation for the ... thing

[hollow knot]

yeah ofc you can

[outer stream]

can u write it without using the summation

maybe i can understand it better

appreciate it

[hollow knot]

alr

Given: $a_{n+2} = a_{n+1} + a_n, a_0=0, a_1=1$

TP $a_{n+2} = 1 + a_0 + a_1 + ... a_n$ :

Let's first check for a few values of n:

if $n = 0$:

$a_2 = a_0 + 1$

$a_1 = a_0 + 1$

$1 = 1$

now for $n=1$:

$a_3 = a_0 + a_1 + 1$

$a_2 + a_1 = a_0 + a_1 + 1$

$1 + 1 = 0 + 1 + 1$

Now, let's assume the following is true for $n=k$:

So, $a_{k+2} = 1 + a_0 + a_1 + ... + a_k$

Now, let's prove it for $n=k+1$:

$a_{k+3} = 1 + a_0 + a_1 + ... a_k + a_{k+1}$

$a_{k+2} + a{k+1} = 1 + a_0 + a_1 + ... a_k + a_{k+1}$

$a_{k+2} = 1 + a_0 + a_1 + ... a_k$

The above is what we assumed to be true

As we derived the above from trying to prove for $n=k+1$

By the hypothesis of induction

hence proved.

[shrewd moat]

Inverse Cupid

[outer stream]

I understand it now

You have no idea how much this means to me

Thank you man

thank you

+close