SOLUTION: Insert the indicated number of arithmetic means between the two given numbers Three; a+2 and 5a-10 So it looks like this a+2 _ _ _ 5a-10 Find the three terms between

Algebra ->  Sequences-and-series -> SOLUTION: Insert the indicated number of arithmetic means between the two given numbers Three; a+2 and 5a-10 So it looks like this a+2 _ _ _ 5a-10 Find the three terms between      Log On


   

Question 1039313: Insert the indicated number of arithmetic means between the two given numbers
Three; a+2 and 5a-10
So it looks like this
a+2 _ _ _ 5a-10
Find the three terms between them
Thank you please help me
Please with solution so i can study it myself
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
ONE WAY TO GET TO THE ANSWER (Probably the expected way):
When you fill in the blanks, the five numbers (or expressions) form an arithmetic sequence.
a%5B1%5D=a%2B2 is the firs term of the sequence.
a%5B5%5D=5a-10 is the fifth element of the sequence.
In an arithmetic sequence each term is the one before plus a common difference, d .
In formulas, that is written as
a%5Bn%2B1%5D=a%5Bn%5D%2Bd , or a%5Bn%5D=a%5Bn-1%5D%2Bd .
As a consequence, any term is related to the first term, and the common difference by the formula
a%5Bn%5D=a%5B1%5D%2B%28n-1%29%2Ad
So, for n=5 , a%5B5%5D=a%5B1%5D%2B%285-1%29%2Ad <---> a%5B5%5D=a%5B1%5D%2B4d
Since we know a%5B1%5D=a%2B2 and a%5B5%5D=5a-10 , we can substitute those expressions and solve for d :
5a-10=a%2B2%2B4d ---> 5a-10-a-2=a%2B2-a-2%2B4d ---> 4a-12=4d ---> %284a-12%29%2F4=d ---> d=a-3 .
Now, we can find the missing terms:
The second term is
a%5B2%5D=a%5B1%5D%2Bd=a%2B2%2Ba-3=highlight%282a-1%29 .
The third term is
a%5B3%5D=a%5B2%5D%2Bd=2a-1%2Ba-3=highlight%283a-4%29 .
The fourth term is
a%5B4%5D=a%5B3%5D%2Bd=3a-4%2Ba-3=highlight%284a-7%29 .

ANOTHER WAY (This works only for sequences with an odd number of terms):
When three numbers are in an arithmetic sequence,
the middle term is the average of the first and last terms.
The same thing happens when in the sequence there are
5 terms, 7 terms, 9 terms, 11 terms, and so on.
So, the middle (third) term is
%28%28a%2B2%29%2B%285a-10%29%29%2F2%2B%286a-8%29%2F2=highlight%283a-4%29 .
The second number, the one between a%2B2 and 3a-4 is
%28%28a%2B2%29%2B%283a-4%29%29%2F2=%284a-2%29%2F2=highlight%282a-1%29 .
The fourth number, the one between 3a-4 and 5a-10 is
%28%283a-4%29%2B%285a-10%29%29%2F2=%288a-14%29%2F2=highlight%284a-7%29 .