document.write( "Question 957473: if \"n\" arithmetic means were inserted between 75 and 19 such that the sum of the third and the fourth means to the sum of the means whose order are (n-3) and (n-4) equals 61: 37 evaluate n. .... I know that n equals 13 but i don't know how \n" ); document.write( "

Algebra.Com's Answer #585094 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
The $\"n\"$ arithmetic means, $\"a%5B1%5D\"$ , $\"a%5B2%5D\"$ , $\"a%5B1%5D\"$ , ..., $\"a%5Bn%5D\"$ ,
\n" ); document.write( "are equally spaced by a common difference $\"d\"$ , forming an arithmetic sequence (or arithmetic progression):
\n" ); document.write( " $\"a%5B0%5D=75\"$ ,
\n" ); document.write( " $\"a%5B1%5D=75-d\"$ ,
\n" ); document.write( " $\"a%5B2%5D=75-2d\"$ ,
\n" ); document.write( " $\"a%5B3%5D=75-3d\"$ ,
\n" ); document.write( " $\"a%5B4%5D=75-4d\"$ ,
\n" ); document.write( "................... ,
\n" ); document.write( " $\"a%5Bn%5D=75-nd\"$ , and
\n" ); document.write( " $\"19=75-%28n%2B1%29d\"$ .
\n" ); document.write( "We do not know $\"n\"$ , and we do not know $\"d\"$ ,
\n" ); document.write( "but we have one equation relating them:
\n" ); document.write( " $\"19=75-%28n%2B1%29d\"$ <---> $\"%28n%2B1%29d=75-19\"$ <---> $\"%28n%2B1%29d=56\"$ .
\n" ); document.write( "We just need another equation.
\n" ); document.write( "From the given ratio we need to get the other equation.
\n" ); document.write( "The easiest way I could think is to look at that arithmetic sequence/progression in reverse.
\n" ); document.write( "Starting from the other end
\n" ); document.write( " $\"a%5Bn%5D=19%2Bd\"$ ,
\n" ); document.write( " $\"a%5Bn-1%5D=19%2B2d\"$ ,
\n" ); document.write( " $\"a%5Bn-2%5D=19%2B3d\"$ ,
\n" ); document.write( " $\"a%5Bn-3%5D=19%2B4d\"$ ,
\n" ); document.write( " $\"a%5Bn-4%5D=19%2B5d\"$ ,
\n" ); document.write( "and so on.
\n" ); document.write( "So, $\"a%5Bn-3%5D%2Ba%5Bn-4%5D=%2819%2B4d%29%2B%2819%2B5d%29=38%2B9d\"$ , and
\n" ); document.write( "we knew that $\"a%5B3%5D=75-3d\"$ and $\"a%5B4%5D=75-4d\"$ , so
\n" ); document.write( " $\"a%5B3%5D%2Ba%5B4%5D=%2875-3d%29%2B%2875-4d%29=150-7d\"$ .
\n" ); document.write( "The ratio of those sums is
\n" ); document.write( " $\"%28150-7d%29%2F%2838%2B9d%29=61%2F37\"$
\n" ); document.write( "Solving that equation for $\"d\"$ we get
\n" ); document.write( " $\"%28150-7d%29%2F%2838%2B9d%29=61%2F37\"$
\n" ); document.write( " $\"37%28150-7d%29=61%2838%2B9d%29\"$
\n" ); document.write( " $\"37-150-37%2A7d=61%2A38%2B61%2A9d\"$
\n" ); document.write( " $\"5550-259d=2318%2B549d\"$
\n" ); document.write( " $\"5550-2318=259d%2B549d\"$
\n" ); document.write( " $\"3232=808d\"$
\n" ); document.write( " $\"3232%2F808=d\"$ ---> $\"d=4\"$
\n" ); document.write( "Now, we plug that value for $\"d\"$ into the equation
\n" ); document.write( " $\"%28n%2B1%29d=56\"$ that we had, and get
\n" ); document.write( " $\"%28n%2B1%29%2A4=56\"$ <--> $\"n%2B1=56%2F4\"$ <--> $\"n%2B1=14\"$ <--> $\"n=14-1\"$ <--> $\"highlight%28n=13%29\"$ \n" ); document.write( "

\n" );