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The sum of the 1st 8 term of an A.P is 60 and the sum of the next six terms is 108. Find the first term and common difference.
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The sum of the nth term of an ap is given by:
Sn = n/2 (2a + (n - 1)d)
where a is the first term and d is the common difference.
for the first 8th term, S8 = 8/2 (2a + (8 - 1)d) = 60
4 (2a + 7d) = 60
divide both sides by 4
2a + 7d = 15 ---------- eqn(1)
fro the firest 6th term, S6 = 6/2 (2a + (6 - 1)d) = 108
3 (2a + 5d) = 108
divide both sides by 3
2a + 5d = 36 -------------- eqn(2)
eqn(1) and eqn(2) are simultaneous equations and can be solved by elimination method.
Subtract eqn(2) from eqn(1),
(2a - 2a) + (7d - 5d) = 15 - 36
2d = -21
divide both sides by 2
d = -21/2
Substitute d = -21/2 into eqn(1)
2a + 7(-21/2) = 15
2a - 147/2 = 15
add 147/2 to both sides
2a = 15 + 147/2
2a = 177/2
divide both sides by 4
a = 177/4
= 44.25
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