### The sum of the first 9terms is 72 and the sum of the next 4terms...

The sum of the first 9terms is 72 and the sum of the next 4terms is 71 find the AP?

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S(n) = Sum of ' n ' terms of an arithmetic sequence
n = number of terms
a = First term
a[n] = nth term
d = Common difference between terms
S(n) = ½ * n[ (2a + (n − 1)d ]
S(9) = 72 = ½ * 9[ 2a + (9 − 1)d ]
(72 * 2)/9 = 2a + 8d
2a + 8d = 16
a + 4d = 8
4d = 8 − a
d = (8 − a)/4 . . . . . . 1st equation
S(13) = ½ * 13[ 2a + (13 − 1)d ]
S(13) = ½ * 13[ 2a + 12d ]
S(13) = 13(a + 6d)
S(13) = 13a + 78d
S(13) − S(9) = 71
S(13) − 72 = 71
S(13) = 143
S(13) = 13a + 78d
143 = 13a + 78d
11 = a + 6d . . . . . 2nd equation
The first term:
From 1st equation , d = (8 − a)/4
11 = a + 6(8 − a)/4
11 = a + (48 − 6a)/4
11 = a + 12 − 3a/2
3a/2 − a = 12 − 11
(3a − a)/2 = 1
a/2 = 1
[ a = 2 ]
The common difference :
Again from first equation d = (8 − a)/4
d = (8 − 2)/4
d = 6/4
[ d = 3/2 ]
The A.P.
a[n] = a + (n − 1)d
a[n] = 2 + (n − 1)3/2
a[n] = 2 + 3n/2 − 3/2
a[n] = (3n + 4 − 3)/2
a[n] = (3n + 1)/2
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