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(10.) The sum of the 6th and 8th terms of an

arithmetic progression is 142. If the fourth
term is 49, find the first term, the common
difference and the sum of the first seven
terms of the progression.

User Justyn
by
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1 Answer

26 votes
26 votes

Answer:

  • first term: 27
  • common difference: 7 1/3
  • sum of 7 terms: 343

Explanation:

The general term of an arithmetic sequence is ...

an = a1 + d(n -1)

Using the given information, we can write two equations in a1 and d:

a6 +a8 = 142 = (a1 +d(6 -1)) +(a1 +d(8 -1)) = 2a1 +12d

a4 = 49 = a1 +d(4 -1) = a1 +3d

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Subtracting twice the second equation from the first, we get ...

(2a1 +12d) -2(a1 +3d) = (142) -2(49)

6d = 44

d = 44/6 = 22/3 = 7 1/3 . . . the common difference

Subtracting the first equation from 4 times the second gives ...

4(a1 +3d) -(2a1 +12d) = 4(49) -142

2a1 = 54

a1 = 54/2 = 27 . . . the first term

_____

The sum of the first n terms of the progression is ...

Sn = (2·a1 +d(n -1))(n/2)

The sum of the first 7 terms is ...

S7 = (2·27 +22/3(7 -1))(7/2) = (54 +44)(7/2) = 343 . . . sum of 7 terms

User MS Berends
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