Question

help please

Answer 1

@Vocaloid

Answer 2

n = the number of terms in the series

since your start at n = 1 and end at n = 32 our n value is 32

for a1, this is the first term of the sequence, you will take the rule 4n + 1 and plug in n = 1 to find the first term

for an, this is the last term of the sequence, take the rule 4n + 1 and plug in n = 32 to find the last term

after that you will plug in the values into sum = (n/2)(a1 + an)

lmk if you're having some trouble

Answer 3

how do i figure out what the next 18 terms?

Answer 4

well, let's start with n = 1

4(n+1) = 4(1+1) = 8

now n = 2:
4(n+1) = 4(2+1) = 12

now n = 3:
4(n+1) = 4(3+1) = 16, etc. notice how it's increasing by 4 each time

so if they want you to list out all 18 terms you would start with 8 and go up to n = 18

if they ~just~ want the 18th term then you could just plug in n = 18 into 4n + 1

Answer 5

oh, so i just have to add it, and change a digit until 21, and then add all the stuff together?

Answer 6

@Vocaloid okay, so i did it, and i got 2144, is that right?

Answer 7

for the sum? yeah 2144 is good, well done

Answer 8

nice, i got another question real quick, can you help too?

Answer 9

sure

Answer 10

What is the sum of the first 21 terms of the arithmetic series?

−5+(−3)+(−1)+1+...

Answer 11

well same process as before we just need to figure out what the rule is

it increases by 2 each time and starts with -5

so we start with an = 2n + (b) and to figure out what b is we start with the first term, n = 1 and a1 = -5

2(1) + b = -5

so b = -7

giving is the sequence rule an = 2n - 7

Answer 12

now, we simply apply the formula from earlier:

a1 = -5
an = 2n - 7, plug in n = 21 to get the an term since we have 21 terms
n = 21 (the number of terms)

sum = (n/2)(a1+an) = ?

Answer 13

126

Answer 14

hm. I got something a little diff.

an = 2n - 7 = 2)21)-7 = 35

so sum = (21/2)(-5+35) = 315 = your sol'n

Answer 15

so it's 315?

Answer 16

yes

Answer 17

can you help with some series and applications?

Answer 18

I'll give it a shot

Answer 19

What is the sum of the first 7 terms of the series −4+8−16+32−... ?

Answer 20

@Vocaloid

Answer 21

well it seems like a geometric series, any ideas on how to write the series rule? as a hint, what is each consecutive term being multiplied by?

Answer 22

you can find this out by picking any term and dividing by the term before it

so if we pick 8 and divide that by -4 we get -2 as the geometric ratio

a geometric series rule can be written as

(first term)(geometric ratio)^(n-1)

giving us

(-4)(-2)^(n-1) as the rule

then we can use the geometric sum formula

Answer 23

where a is the first term (-4), r is the ratio (-2), and n = number of terms (7)

gtg on lunch break

Answer 24

kk

Answer 25

so, made any progress yet?

a(r^n-1)/(r-1) = ?

where a = (-4), r = -2, n = 7?

Answer 26

yeah, i got past that after you left, forgot what i wrote, sadly

Answer 27

still want to help some more?

Answer 28

sure

Answer 29

The explicit rule for a sequence is an=9(−5)n−1 .



What is recursive rule for the sequence?

Answer 30

so you just need to find the first term by plugging in n = 1 into 9(−5)n−1 [not sure if there's supposed to be an exponent there? 9(−5)^(n-1)?]

after that you can write the recursive rule as:

a1 = ____ (insert first term here)
an = [common ratio] * \(a_{n-1}\)

(if you need some help identifying the common ratio it's the number being raised to the n-1 exponent)

Answer 31

anyway, if we look at the original formula we have 9(-5)^(n-1)

plugging in n = 1 gives us 9(-5)^0 = 9 as the first term
common ratio = -5 (notice how the -5 is being raised to the n-1 power)

so recursive formula:'

a1 = 9
\(a_{n}\) = (-5)* \(a_{n-1}\)

both of these lines are necessary to specify the formula

Answer 32

still kinda got me stumped

Answer 33

so the recursive formula has two lines:

a1 = ____ the first term
an = [some sort of calculation] * a_n-1, this tells us how to get to the next term from the previous term a_n-1

where a_n-1 is the previous term

according to our calculations, we found that 9 is the first term so we write

a1 = 9

and the common ratio was (-5), this gives us a geometric series where

\(a_{n}\) = (-5)* \(a_{n-1}\), this means "take the previous term and multiply it by -5 to get to the next term"

Answer 34

therefore the complete formula is:

a1 = 9
\(a_{n}\) = (-5)* \(a_{n-1}\)