Answer:
- first term: 27
- common difference: 7 1/3
- sum of 7 terms: 343
Step-by-step 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
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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
