(i) 3, 5, 7, 9, ..... 201
a = 3, d = 5 - 3 = 2, an = 201
an = a +(n -1)d
201 = 3 +2n – 2
N = 100
Now, we have to find 12th term from the last that means,
100th – 11 = 89th term
Then,
a89 = a + (89 – 1)d
= 3 + 88 x 2
= 179
Hence, the 12th term from the end of the A.P. is 179.
(ii) 3, 8, 13, ....., 253
a = 3, d = 8 - 3 = 5
an = 253
a + (n - 1)d = 253
3 + (n - 1)5 = 253
n = 51
Now, we have to find 12th term from the last that means,
51th -11 = 40th term
Then,
a40 = a + (n -1)d
= 3 + 39(5)
= 198
Hence, the 12th term from the end of the A.P. is 198
(iii) 1, 4, 7, 10, ......., 88
a = 1, d = 4 -1 = 3
an = 88
a + (n -1)d = 88
1 + (n -1)3 = 88
n = 30
Now, we have to find 12th term from the last that means,
30th - 11 = 19th term
a19 = a + 18d
= 1 + 18(3)
= 55
Hence, the 12th term from the end of the A.P. is 198.