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The sum of the 1^{st} 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:

S_{n} = n/2 (2a + (n - 1)d)

where a is the first term and d is the common difference.

for the first 8th term, S_{8} = 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, S_{6} = 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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