Question 1184342
.
Insert the indicated number of arithmetic means given the first and last term, show the solution.
1. -5 and 50 indicated number [6]
2. -12 and 12 indicated number [5]
3. 21 and -9 indicated number [4]
~~~~~~~~~~~~~



I will solve parts  (2)  and  (3).

I will show you a simple method and a simple mantra,  which you will be able easy apply in many other cases you can meet.



<U>PART (2)</U>


<pre>
     Between the endpoints, -12 and 12, the distance is 24 units.

     5 inserted arithmetic means divide this distance, this segment [-12,12], in 5+1 = 6 gaps of equal length.

     The length of each such gap is  24/6 = 4 units, and it is the common difference of the arithmetic sequence.

     Hence, the sequence is  -12, -8, -4, 0, 4, 8, 12.

     The arithmetic means which you INSERT are -8, -4, 0, 4 and 8.
</pre>

Solved and completed.



<U>PART (3)</U>


<pre>
     Between the endpoints, -9 and 21, the distance is 30 units.

     4 inserted arithmetic means divide this distance, this segment [-9,21], in 4+1 = 5 gaps of equal length.

     The length of each such gap is  30/5 = 6 units, and it is the common difference of the arithmetic sequence.

     Hence, the sequence is  -9, -3, 3, 9, 15, 21.

     The arithmetic means which you INSERT are -3, 3, 9 and 15.
</pre>

Solved and completed.



/////////////



Do you see the repeating mantra and how it works ?



Now you know the method.  &nbsp;&nbsp;&nbsp;&nbsp;Apply it in thousands other similar problems.