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) About Me  (Show Source):
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.
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

    a%5B1%5D + a%5B2%5D + a%5B3%5D + a%5B4%5D = 26,    (1)

    a%5B5%5D + a%5B6%5D + a%5B7%5D + a%5B8%5D = 74.    (2)


It is the same as

    4%2Aa%5B1%5D + (1+2+3)d   = 26,     

    4%2Aa%5B1%5D + (4+5+6+7)d = 74,

or

    4%2Aa%5B1%5D +  6d = 26,   (3)

    4%2Aa%5B1%5D + 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)

    4%2Aa%5B1%5D = 26 - 6*3 = 26 - 18 = 8,

    a%5B1%5D = 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) About Me  (Show Source):
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The sum of the first four terms of a linear sequence A.p is 26 
Let a = first term and d = common difference

a%2B%28a%2Bd%29%2B%28a%2B2d%29%2B%28a%2B3d%29=26
4a%2B6d=26
2a%2B3d=13
and that of the next four term is 74.
%28a%2B4d%29%2B%28a%2B5d%29%2B%28a%2B6d%29%2B%28a%2B7d%29=74%29
4a%2B22d=74
2a%2B11d=37
find the value of the first term ,the common difference
Solve the system:

system%282a%2B3d=13%2C+2a%2B11d=37%29

Edwin

Answer by greenestamps(13200) About Me  (Show Source):
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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:

16d=48
d=3

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:

4%28%28a%2Ba%2B9%29%2F2%29=26
4%282a%2B9%29=52
2a%2B9=13
2a=4
a=2

ANSWERS: first term a=2; common difference 3

Answer by MathTherapy(10552) About Me  (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.: matrix%281%2C3%2C+S%5Bn%5D%2C+%22=%22%2C+%28n%2F2%29%282a%5B1%5D+%2B+%28n+-+1%29d%29%29

                      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