8. The 11th term and the 21st term of an A.P. are 16 and 29 respectively, find :
(i) the 1st term and the common difference.
(ii) the 34th term
(iii) ‘n’ such that tn = 55.
Sol. tn = a + (n – 1) d
∴ t11 = a + (11 – 1) d
16 = a + 10d
∴ a + 10d = 16
∴ a = 16 – 10d .......(i)
∴ t21 = a + (21 – 1) d
29 = a + 20d
∴ a + 20d = 29 ......(ii)
Substituting eq. (i) in eq. (ii)
∴ a + 20d = 29 ......(ii)
∴ 16 – 10d + 20d = 29
∴ 10d = 29 – 16
∴ 10d = 13
∴ d = 13/10
∴ d = 1.3
Substituting d = 1.3 in eq. (i)
a = 16 – 10d .......(i)
∴ a = 16 – 10 (1.3)
∴ a = 16 – 13
∴ a = 3
The first term is 3 and the common difference is 1.3
(ii) the 34th term
Sol. tn = a + (n – 1) d
t34 = a + (34 – 1) d
∴ t34 = 3 + 33 (1.3)
∴ t34= 3 + 42.9
∴ t34 = 45.9
∴ Thirty fourth term of A.P. is 45.9.
(iii) ‘n’ such that tn = 55. (1 mark)
Sol. tn = a + (n – 1) d
∴ 55 = 3 + (n – 1) d
∴ 55 = 3 + (n – 1) 1.3
∴ 55 = 3 + 1.3n – 1.3
∴ 55 = 1.7 + 1.3n
∴ 53.3 = 1.3n
∴ n = 53.3/1.3
∴ n = 53.3/1.3 × 10/10
∴ n = 533/13
∴ n = 41
