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Question

The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.

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Solution

The nth term of an A.P is given by
an=a1+(n1)d
4th term of A.P is given by
a4=a1+(41)d
a4=a1+3d...eq(1)

8th term of A.P is given by
a8=a1+(81)d
a8=a1+7d...eq(2)

given that sum of 4th and 8th term is 24 a4+a8=24
from eq(1) and eq(2)
a1+3d+a1+7d=24
2a1+10d=24...eq(3)

6th term of A.P is given by
a6=a1+(61)d
a6=a1+5d...eq(4)

10th term of A.P is given by
a10=a1+(101)d
a10=a1+9d...eq(5)

given that sum of 6th and 10th term is 34 a6+a10=34
from eq(4) and eq(5)
a1+5d+a1+9d=34
2a1+14d=34...eq(6)

substract eq(3) by eq(6) we get
2a12a1+14d10d=3424
4d=10
d=104
d=52.....eq(7) common difference.

put value of d=2.5 in eq(3) we get
2a1+10×2.5=24
2a1+25=24
2a1=2425
2a1=1
a1=12= first term of A.P

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