find the missing terms of the arithmetic sequence
2, __, __, __, -0.4

Question

find the missing terms of the arithmetic sequence 
2, __, __, __, -0.4

anonymous · Answer

please help @whpalmer4

whpalmer4 · Answer

Okay, which term numbers do we know?  We know $a_1$, and $a_?$

anonymous · Answer

ok is A1=2 and A=-0.4??

anonymous · Answer

sorry read it wrong it is A5

whpalmer4 · Answer

what is the term number of -0.4?  is it $a_1, a_2, a_3, a_4,a_5,a_6,a_7,a_{9375234}$?

whpalmer4 · Answer

Right, it is $a_5$.  So what's the formula for the $n$th term of an arithmetic sequence?

whpalmer4 · Answer

$$a_n = a_1 ...$$

anonymous · Answer

An=A1+(n-1)(d)    right

whpalmer4 · Answer

Yes.  So we know $a_1$ and $a_5$.  Can you use that information to find $d$?

anonymous · Answer

i think so what if we use A1=2   and  n=5   and we leave d as d right

whpalmer4 · Answer

go for it!

anonymous · Answer

well what i have is
$$a _{n}=6(d)$$
what do i do now

whpalmer4 · Answer

no, you don't have that...

you started with $$a_n= a_1 + (n-1)d$$then you plugged in n = 5
$$a_5 = a_1 + (5-1)d$$$$a_5 = a_1+4d$$$$a_5 = 2+4d$$$$-0.4 = 2+4d$$

You just didn't realize that is what you did ;-)

anonymous · Answer

oh ok so what do we do next??

whpalmer4 · Answer

Solve $$-0.4 = 2 + 4d$$ for $d$, silly :-)

anonymous · Answer

oh im retarded lol!!!! -_-
the answer is d=-.6

whpalmer4 · Answer

yes.  I prefer to always put a 0 in front of the decimal point, just to make it a bit easier to see (and not forget).

So what's the whole formula, and what are the missing terms?

anonymous · Answer

ok the formula is 
$$a_{n}=2+(n-1)(-0.6)$$
and the missing terms are $$a_{2}, a_{3}, a_{4}$$

whpalmer4 · Answer

right.  what are the values of $a_2, a_3, a_4$?

anonymous · Answer

$$a_{2}=1.4, a_{3}=.8, a_{4}=.2$$

whpalmer4 · Answer

very good.

anonymous · Answer

thank you so much!!!!

whpalmer4 · Answer

the key here was to realize that even though we only knew two terms, we really knew 3 of the 4 things in the equation.

whpalmer4 · Answer

Now, how about I bend your mind a bit with a problem of my own, which you might well encounter in your homework or on a test, but haven't asked me yet so far...

whpalmer4 · Answer

You know two terms in an arithmetic sequence.  One of them, $a_4 = 9$.  The other one, $a_{10}=21$.  What is the formula for the sequence?

anonymous · Answer

i have no clue!!!!! 
you just made my face go 0.o

whpalmer4 · Answer

it's not as hard as it seems...

anonymous · Answer

i dont know how to start

whpalmer4 · Answer

brb, nature calls

anonymous · Answer

lol ok

whpalmer4 · Answer

okay, got the solution yet? :-)

anonymous · Answer

no i dont

whpalmer4 · Answer

I'll show you two ways you could work it out, the "in your head" way, and the more formal way.  which do you want first?

anonymous · Answer

um formal

whpalmer4 · Answer

Okay.  We know two terms.  We don't know $a_1$, and we don't know $d$.

$$a_4 = 9$$$$a_{10}=21$$
Let's write the equations:
$$a_4 = a_1+(n-1)d = a_1+(4-1)d = a_1+3d$$$$a_{10} = a_1+(n-1)d = a_1+(10-1)d = a_1+9d$$
Think about that for a minute, and tell me if it gives you the answer...

whpalmer4 · Answer

remember, we know the values of $a_4$ and $a_{10}$...

anonymous · Answer

umm not sure will d=6

whpalmer4 · Answer

ooh....you're on the right track...

whpalmer4 · Answer

but going in the wrong direction :-)

anonymous · Answer

ok what do i do

whpalmer4 · Answer

Let's write the equations with those values plugged in:

$$21=a_1+6d$$$$\,\,\,\,9 = a_1+3d$$-------------

What if we subtract those two equations from each other?  Straight down the columns...

anonymous · Answer

12=6d

whpalmer4 · Answer

wait, stop!  I typed the wrong things...

$$21 = a_1 + 9d$$$$\,\,\,\,9 = a_1+3d$$
-----------------------------

anonymous · Answer

so then when we divide we will get d=2

whpalmer4 · Answer

yes, you got the right answer, even though I typed the wrong equation :-)

whpalmer4 · Answer

a case of two wrongs making a right :-)

whpalmer4 · Answer

Next time someone tells you that two wrongs don't make a right, you can tell them you've seen at least one case where they did :-)

anonymous · Answer

i knew what you were trying to say so i used the original equations to find the answer

whpalmer4 · Answer

Ah, good for you!  

So, the "in your head" way is to observe that $a_{10}$ is 6 terms on from $a_{4}$ in the sequence, so the difference between the two must be 6 * the common difference.  21 - 9 = 12, and 12/6 = 2, so the common difference is 2.

anonymous · Answer

i like that way better it is easier lol so hey will you help me with one more question if i make it a new question

whpalmer4 · Answer

Kind of neat how that works, isn't it?  At first, your reaction is "how could we possibly figure that out?" and then when you do, it's more like "how could we possibly not have realized that right away?" :-)

whpalmer4 · Answer

I'm always happy to help you...