Summation Notation

[frank copper]

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[limpid bone]

first term is 5

and common ratio is 5

[latent vale]

$a_n = a_0+(n-1)d$

[narrow crest]

xNiden

[limpid bone]

this is geometric

so its

$a_n = ar^{n-1}$

[narrow crest]

No Lifer #GTFOanemia

[limpid bone]

now you wish to find a partial sum ryt

[latent vale]

yes

[limpid bone]

remember the formula to find the sum up till n terms of a geometric sequence

[latent vale]

$a_n = 5(5)^{n-1}$

[narrow crest]

xNiden

[latent vale]

,calc 5(5)^{1-1}+5(5)^{2-1}+5(5)^{3-1}+5(5)^{4-1}+5(5)^{5-1}+5(5)^{6-1}

[narrow crest]

The following error occured while calculating:
Error: Symbol or string expected as object key (char 7)

[limpid bone]

or do I make you derive it lol

yeah, you're deriving it

alr

let's start with a loose notation of a geometric series

$a + ar + ar^2 + \dots + ar^{n-1}$

[narrow crest]

No Lifer #GTFOanemia

[limpid bone]

let's take a common

$a(1 + r + r^2 + \dots r^{n-1})$

[narrow crest]

No Lifer #GTFOanemia

[limpid bone]

now, there's an identity which states that

$(x-a)(a^{k-1}+xa^{k-2}+x^2a^{k-3}+\dots+x^{k-1}) = \frac{x^k - a^k}{x-a}$

[narrow crest]

No Lifer #GTFOanemia

[latent vale]

$S_n = \frac{a_1(1-r^n)}{1-r}$

[narrow crest]

xNiden

[limpid bone]

yep

use that

[latent vale]

$S_n = \frac{5(1-5^n)}{1-5}$

[narrow crest]

xNiden

[limpid bone]

oof

[wary gorge]

No

[limpid bone]

1st mistake

$S_n = \frac{5(5^n - 1)}{4}$

[narrow crest]

No Lifer #GTFOanemia

[limpid bone]

continue from here

[latent vale]

do we not distribute?

wait

[limpid bone]

nope

no need

now just sub 6 everywhere

for n

[latent vale]

why did you swap 5^n with 1?

wouldn't it be -5^n?

[limpid bone]

bottom

we had 1-5

I made it 5-1
=4

[latent vale]

no i mean numerator

[latent vale]

this would just be $S_n = \frac{5(1-5^n)}{1-5}$

[limpid bone]

I multiplied both numerator and denominator by -1

yeah that's correct

[narrow crest]

xNiden

[latent vale]

oh

then where did the -1 go?

doesn't that mean it would be $S_n = \frac{-5(-1+5^n)}{-1+5}$

[wary gorge]

-1 times the numerator and it goes into ( )

[narrow crest]

xNiden

[limpid bone]

$\frac{5(1-5^n)}{1-5} = \frac{5(5^n-1)}{5-1}$

[narrow crest]

No Lifer #GTFOanemia

[latent vale]

oh both being negative

making it positive

mb

[limpid bone]

yep

[latent vale]

ok i get it now

[limpid bone]

alr

[latent vale]

i was confused sorry about that

[limpid bone]

,calc 5^6

[narrow crest]

Result:

15625

[latent vale]

thanks for the clarification

[limpid bone]

alr

now I think you can solve

you have 5^6 now

[latent vale]

so then i just put in 1-6 for n?

[limpid bone]

no

just put 6 lol

[latent vale]

so $\frac{5(5^6-1)}{5-1}$

[narrow crest]

xNiden

[limpid bone]

yes

[latent vale]

,calc (5(5^6-1))/4

[narrow crest]

Result:

19530

[wary gorge]

ay this is the same as calculating all of the values itself right

[latent vale]

ah i see

[latent vale]

yeah but quicker

[limpid bone]

lol

oh well

[wary gorge]

yeah

I never did summation for these

[limpid bone]

oof

[latent vale]

$\frac{5(5^6-1)}{5-1}$

[narrow crest]

xNiden

[latent vale]

ima do it manually

[limpid bone]

try deriving their summation formula

[limpid bone]

alr

[wary gorge]

But it’s just general rules you apply

yeah I saw you explaining it

[limpid bone]

alr then

[wary gorge]

dunno about the identity tho

I lost the plot

[latent vale]

$\frac{5(15624)}{4}$

[wary gorge]

around there

[limpid bone]

the identity I used can be proved by the binomial theorem

[wary gorge]

Right

[narrow crest]

xNiden

[limpid bone]

but that's taught after geometric series I think

[wary gorge]

Wait

[limpid bone]

first divide

[latent vale]

$\frac{78125}{4}$

[narrow crest]

xNiden

[limpid bone]

bruh

[wary gorge]

I’ve had that one

[limpid bone]

no

[wary gorge]

binomial theorem

[limpid bone]

oh damn

hmmm

[latent vale]

$19531$

[narrow crest]

xNiden

[latent vale]

oh

I did it wrong

lol

[limpid bone]

yeah...

[latent vale]

wait where did i mess up

[wary gorge]

,calc (5/4) x 15624

[narrow crest]

The following error occured while calculating:
Error: Undefined symbol x

[wary gorge]

,calc (5/4) * 15624

[narrow crest]

Result:

19530

[limpid bone]

yeah

that's what you should be getting

[latent vale]

yeah

I was off by 1

hm

[wary gorge]

lol

[limpid bone]

yes 1

dude was off by one

[wary gorge]

how

[latent vale]

why would you divid 5/4 before rather than dividing it all together?

[limpid bone]

anyways, it's a calculation error

[wary gorge]

No it doesn’t matter

you can aslo

[limpid bone]

divide by 4 first
because you'll work w/ smaller numbers that way

[wary gorge]

,calc 5 * 15624

[narrow crest]

Result:

78120

[wary gorge]

,calc 78120 / 4

[narrow crest]

Result:

19530

[wary gorge]

Doesn’t really matter so idk how you got the wrong answer

Let me see

78125

Idk where that came from

[latent vale]

5^6-1 = 15624

i see my mistake

I forgot to -1

that's how i got 1 off

[wary gorge]

Alr

[latent vale]

i did 5^6 * 5

thanks

[wary gorge]

But how does that uh

identity

come from binomial theorem

[limpid bone]

that idk
oof

[wary gorge]

yeah well I only used it to rewrite (a + b)^n

pretty much

[latent vale]

no like

hold on

$\frac{5(15624)}{4}$

so i got this

[narrow crest]

xNiden

[latent vale]

but

in my calculator I did 5*5^6

cause I forgot the -1 which would make up for the 15624

that's why i was off by 1

[wary gorge]

Right

Seems logical

I wouldn’t even have had to learn this

if it wasn’t for my own choice

Perhaps my education system is different

no but like geometric is multiplication right and you had something else I don’t know the English term for it but it was addition

[latent vale]

$a_n$ = $a_1r^{n-1}$

[narrow crest]

xNiden

[latent vale]

$-32768$ = $-2*4^{n-1}$

[narrow crest]

xNiden

[latent vale]

,calc -32768/-2

[narrow crest]

Result:

16384

[latent vale]

$16384 = 4^{n-1}$

[narrow crest]

xNiden

[latent vale]

,calc (-2(1-4^4095))/-1

[narrow crest]

The following error occured while calculating:
Error: Symbol or string expected as object key (char 10)

[latent vale]

,calc (-2(1-4^4095)/-1

[narrow crest]

The following error occured while calculating:
Error: Parenthesis ) expected (char 17)

[latent vale]

,calc (-2(1-4^4095))/-1

[narrow crest]

Result:

-Infinity

[latent vale]

,calc (-2(1-4^4095))/-1

[narrow crest]

Result:

-Infinity

[latent vale]

hm

+close