In an decreasing AP, the sum of all its terms except the first term is equal to -36 and the sum of all its terms except for the last term is zero and the difference of the tenth and the 6th term is equal to -16 then find the first term of the series?

Question : In an decreasing AP, the sum of all its terms except the first term is equal to -36 and the sum of all its terms except for the last term is zero and the difference of the tenth and the 6th term is equal to -16 then find the first term of the series?

Doubt by Muskan

Solution :

Long Method :

a2 + a3 + . . . + an = -36 — (1)

a1 + a2 + a3 + . . .+ an-1 = 0 — (2)

a10 - a6 = -16
a + 9d - (a + 5d) = -16
a + 9d - a - 5d = -16
4d = -16
d = -16/4
d = - 4 — (3)

Subtracting equation (2) from (1)

-a1 + an = -36
-a + an = -36
an = -36 + a — (4)

a + (n-1)d = -36 + a
(n-1)d = -36
(n-1)(-4) = -36 [d =-4]
n-1 = -36/-4
n-1 = 9
n = 9+1
n = 10 — (5)
sn = n/2(a + an)
-36 + a = 10/2 [(a -36 + a)] [using eq 4 and 5]
-36 + a = 5 (2a - 36)
-36 + a = 10a - 180
-36 + 180 = 10a -a
144 = 9a
a = 144/9
a = 16

Shortcut Method :
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In an decreasing AP, the sum of all its terms except the first term is equal to -36 and the sum of all its terms except for the last term is zero and the difference of the tenth and the 6th term is equal to -16 then find the first term of the series?