Question

i have a limited amount of notes for my final and i don't understand sequences or series…any tips for whats most important to know? thank you

Answer 1

based on my understanding of series and sequences you need to memorize these formulas

nth term of arithmetic sequence
\[a_n = a_1 + (n-1)d\]

sum of arithmetic sequence
\[S_n = \frac{n(a_n +a_1)}{2}\]

nth term of geometric sequence
\[a_n = a_1 r^{n-1}\]

sadly i cant remember the formula for sum of geometric sequence

Answer 2

sum=a/1-r

Answer 3

r<1

Answer 4

is that a_n or a_1?

Answer 5

thank you i know the formulas but I'm having a hard time understanding which one to use at what time

Answer 6

and the \[s^{n}\] i don't really get

Answer 7

arithmetic sequence is when \[a_2 - a_1 = a_4 - a_3\]
that means there is a " common difference" in the terms

geometric sequence is \[a_2 \div a_1 = a_4 \div a_3\]
that means there is a "common quotient" in the terms

Answer 8

\[s_n\]
that means sum of n terms

Answer 9

let's take an example. you have the terms

1, 2, 3, 4, 5, 6

this is an arithmetic sequence because the next term is just ADDING 1 from the previous term (keyword adding)

1, 2, 4, 8, 16, 32, 64

this is geometric because the next term is MULTIPLYING 2 to the previous term (keynote multiplying)

Answer 10

ok i get that now, so when will i use the \[s _{n}\]

Answer 11

let's say you have

1, 2, ,3,....

find the sum of the first 10 terms

from here you can see this is an arithmetic sequence so you use the formulas for it.

but first you need to solve for the 10th term

\[a_{10} = 1 + (10 - 1)(1)\]
this is because a_1 is 1 because it's the first term
10 is n because there are 10 terms
and 1 is d because you're adding 1 to each term
\[a_{10} = 1 + 9\]
\[a_{10} = 10\]
so now we know the tenth term is 10
so now we use \[s_n = \frac{10 (10 + 1)}{2}\]
this is because 10 is n because there are 10 terms
a_n is 10 because it's the last term
a_1 is 1 because it's the first term
\[s_n = \frac{10(11}{2}\]
\[s_n = 5 (11)\]
\[s_n = 55\]
so the sum of the first 10 terms is 55

Answer 12

it's usually explicitly stated you need to find the sum

Answer 13

you use sn if it's looking for sum

you use an if it's just looking for a term

Answer 14

sometimes you need to analyze what's given to you and react accordingly

Answer 15

for example you are given the sum of the numbers, the number of terms and the first term. you're asked to look for the last term, you need to use some algebra

Answer 16

well i need to go now so i wish you luck

Answer 17

thank you so much! i get it now.

Answer 18

you are most welcome :D