SOLUTION: The sum of the first four terms of a linear sequence (A.P) is 26 and that of the next four terms is 74.find the values of (i)the first term (ii)the common difference.

Question 1189392: The sum of the first four terms of a linear sequence (A.P) is 26 and that of the next four terms is 74.find the values of (i)the first term (ii)the common difference.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52812)   (Show Source): You can put this solution on YOUR website!
.

The sum of the first 4 terms is + + + . The sum of the next 4 terms is + + + . Each difference , , and is equal to 4d, where d is the common difference of the AP. Therefore, 4*(4d) = 74 - 26, or 16d = 48; hence, d = 48/16 = 3. Next, 26 = 4a + (1+2+3)d = 4a + 6*3 = 4a + 18, which implies 4a = 26 - 18 = 8. Answer. The first term of the AP is 8/4 = 2; the common difference is 3.

Solved.

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On arithmetic progressions, see the lessons
    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
    - One characteristic property of arithmetic progressions
    - Solved problems on arithmetic progressions
    - Calculating partial sums of arithmetic progressions
    - Finding number of terms of an arithmetic progression
    - Advanced problems on arithmetic progressions
    - Problems on arithmetic progressions solved MENTALLY
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Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!

I will provide a response that solves the problem a bit more informally than the other tutor....

Each of terms 5-8 is equal to the corresponding term among the first four terms, plus 4 times the common difference. (The 5th term is 4 terms after the 1st, so it is equal to the first term plus 4 times the common difference; likewise for the other pairs of corresponding terms.)

So the difference between the sum of the first four terms and the second four terms is 4*4=16 times the common difference.

The difference between those sums is 48, so the common difference in the sequence is 48/16=3.

Using a for the first term, and using the common difference of 3, the first four terms are

a, a+3, a+6, a+9

The sum of those first four terms is 26:

(a)+(a+3)+(a+6)+(a+9) = 26
4a+18=26
4a=8
a=2

ANSWERS: The first term is 2; the common difference is 3

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