series fm

[abstract wagon]

please can someone help me with part c but only the highlighted bit. Idk why it’s 2rt3 and not 5rt3. Thank youuu

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[west shard]

its not 5rt3 because when you find the the sum from 5 to 1

you'd need to sub in 5/3, not 5 bc if you look at the summation, you have 3k on the top

and since k is a natural number (counting number) you cant do that

so what you want to do is find the sum from 6 to 1

then subtract the 6th term

because of that 3k on the top you can only find the sums from term 1 to 3, term 1 to 6, term 1 to 9 etc

so the sum from 6 to 1 would be -2rt3

then for the 6th term you just sub in r = 6 into the rtan60r thing and yeah

[abstract wagon]

I’m so confused 😭

[abstract wagon]

idk how you got that 😔 sorry I’m acc so dumb

[abstract wagon]

so the sum from 6 to 1 would be -6rt3

[abstract wagon]

and the 6th term is just 0

[west shard]

yh its tough to grasp icl but when u do its fine

[west shard]

so you see the rtan60r sum

it starts from 1 and ends at 3k

you know that sum is equal to -krt3

so when k = 1, the sum is -rt3, when k = 2 the sum is -2rt3

-rt3 is the sum from 1 to 3k where k = 1, so from r = 1 to r = 3

and -2rt3 is the sum from r = 1 to r = 3k where k = 2, in other words the sum from r = 1 to r = 6

[west shard]

yeah

so now if you want to find the sum from r = 3n to r = 6

you need to do sum of r = 3n to r = 1 minus r = 6 to r = 1

this first part you just use the result in part b, so -nrt3

now for the second part

[west shard]

it's all the stuff i was on about from here onwards

and you get -2rt3

does that make sense or nah?

[abstract wagon]

wait so why is the sum of r=3n to r=1 -nrt3

[west shard]

the first few terms are
r = 1: rt3
r = 2: -2rt3
r = 3: 0
r = 4: 4rt3
r = 5: -5rt3
r = 6: 0

notice how for the r = 3n terms

so r = 3, r = 6 etc u get 0

then for the first two terms you get -rt3 when you add them

you also get this when adding r = 4 and r = 5

so if two of these pairs i guess give you -2rt3

'n' of these pairs give you -nrt3

[abstract wagon]

OMG THANK YOU SO MUCH

I WAS SO CONFUSED AND YOU ACTUALLY HELPED ME SM

IM SO GRATEFUL TYSMMMM

💗💗

it makes sense now