SOLUTION: The sum of the first four terms of a linear sequence A.p is 26 and that of the next four term is 74. find the value of the first term ,the common difference

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Question 1199651: The sum of the first four terms of a linear sequence A.p is 26 and that of the next four term is 74. find the value of the first term ,the common difference
Found 4 solutions by ikleyn, Edwin McCravy, greenestamps, MathTherapy:
Answer by ikleyn(52788)   (Show Source): You can put this solution on YOUR website!
.
The sum of the first four terms of a linear sequence A.p is 26
and that of the next four term is 74.
find the value of the first term ,the common difference
~~~~~~~~~~~~~~~~~~~~ 

We are given

     +  +  +  = 26,    (1)

     +  +  +  = 74.    (2)


It is the same as

     + (1+2+3)d   = 26,     

     + (4+5+6+7)d = 74,

or

     +  6d = 26,   (3)

     + 22d = 74.   (4)


From (4), subtract (3).  You will get

    22d - 6d = 74 - 26 

       16d   =    48

         d   =    48/16 = 3.


Thus the common difference of this AP is 3.


Next, from (3)

     = 26 - 6*3 = 26 - 18 = 8,

     = 8/4 = 2.


ANSWER.  The first term of the AP is 2;  the common difference is 3.


CHECK.  Use equation (4) to check this answer ON YOUR OWN.

Solved.



Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
The sum of the first four terms of a linear sequence A.p is 26 
Let a = first term and d = common difference
and that of the next four term is 74.



find the value of the first term ,the common difference
Solve the system:



Edwin
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Each number in the second sub-sequence of 4 terms is equal to the corresponding number in the first sub-sequence of 4 terms, plus the common difference 4 times.

So the sum of the 4 terms of the second sub-sequence is 4(4) = 16 times the common difference.

The difference between the sums of the two sub-sequences is 74-26 = 48:




The common difference for the sequence is 3.

The first term of the sequence is a; the fourth term is a+3d = a+9.

The sum of the first four terms of the sequence is 4 times the average of the first and fourth terms; that sum is 26:







ANSWERS: first term a=2; common difference 3
Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
The sum of the first four terms of a linear sequence A.p is 26 and that of the next four term is 74. find the value of the first term ,the common difference
Sum of an A.P.: 

                      First 4 terms:                           ALL 8 terms, including terms 5 - 8:
          
   
                                  13 = 2a1 + 3d ---- eq (i)
                                  25 = 2a1 + 7d ---- eq (ii)
                                  12 = 4d ----- Subtracting eq (i) from eq (ii)
            Common difference, or 

                                  13 = 2a1 + 9 ----- Substituting 3 for d in eq (i)
                              13 - 9 = 2a1 
                                   4 = 2a1 
                  First term, or

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