Arithmetic sequence, geometric sequence, & geometric series

• Jan 23rd 2007, 05:44 AM
fw_mathis
Arithmetic sequence, geometric sequence, & geometric series
Can someone please put me on the right track to solving these problems? If you can provide an example for at least one of each, I would greatly appreciate it. Thanks so much. Please see attachment. ,:( :confused:
• Jan 23rd 2007, 08:20 AM
CaptainBlack
Quote:

Originally Posted by fw_mathis
Can someone please put me on the right track to solving these problems? If you can provide an example for at least one of each, I would greatly appreciate it. Thanks so much. Please see attachment. ,:( :confused:

Quote:

Question 1:
Use the arithmetic sequence of numbers 2, 4, 6, 8, 10… to find the following:
What is d, the difference between any 2 terms?
Why is this a problem?

the differences can be written:

(4-2)=2, (6-4)=2, (8-6)=2, (10-8)=2.

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Question 2:
Using the formula for the nth term of an arithmetic sequence, what is 101st term?

the first term is 2, the second term is 2+2=4, the third term is 2+2+2=6, the n-th term is 2+2+....+2 (with n terms) which is equal to 2n.

Therefore the 101-st tem is 1x101=202.

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Using the formula for the sum of an arithmetic sequence, what is the sum of the first 20 terms?

You need to know the formula for the sum of an arithmetic progression.

S(n)=(1/2)n(2a+(n-1)d)

here first term a=2, common difference d=2 and n=20, so:

S(20)=(1/2)x20x(4+19x2)=420

RonL
• Jan 23rd 2007, 08:47 AM
Soroban
Hello, fw_mathis!

Are you really having trouble with these problems?

Do you know what "difference between any two terms" means?
. . And you're expected know the necessary formulas.
So where exactly is your difficulty?

Quote:

Using the arithmetic sequence of numbers: $\displaystyle 2,\,4,\,6,\,8,\,10,\,\hdots$ to find the following:
What is $\displaystyle d$, the difference between any two consecutive terms?

The difference between 2 and 4 is . . . um . . . 2 ?
The difference between 4 and 6 is . . . er . . . 2 ?

Quote:

Using the formula for the $\displaystyle n^{th}$ term of an arithmetic sequence, what is the $\displaystyle 101^{st}$ term?

The formula for the $\displaystyle n^{th}$ term is: .$\displaystyle a_n\:=\:a + (n-1)d$
. . where $\displaystyle a$ = first term, $\displaystyle d$ = common difference, $\displaystyle n$ = number of terms.

We have: .$\displaystyle a = 2,\:d = 2,\:n = 101$

Therefore: .$\displaystyle a_{101} \;=\;2 + (101-1)2 \;=\;202$

Quote:

Using the formula for the sum of an arithmetic sequence,
. . what is the sum of the first $\displaystyle 20$ terms?

Formula: .$\displaystyle S_n\;=\;\frac{n}{2}\left[2a + (n-1)d\right]$

We have: .$\displaystyle a = 2,\:d = 2,\;n = 20$

Therefore: .$\displaystyle S_{20} \;=\;\frac{20}{2}\left[2(2) + (20-1)2\right] \;=\;420$

I'll let someone else teach you about Geometric Sequences . . .