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4 months ago
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