Question

Write the expression an-ak for the A.P.a,a+d,a+2d,...
Hence, find the common difference of the A.P. for which
(i)11th terms is 5 and 13th term is 79.
(ii)a₁₀-a₅=200
(iii) 20th term is 10 more than the 18th term.

Answer 1

answer : (i) 37 (ii) 40 (iii) 5

here, an = a + (n - 1)d

and ak = a + (k - 1)d

so, an - ak = a + (n - 1)d - {a + (k - 1)d}

= (n - k)d

hence, an - ak = (n - k)d

(i) 11th term = a + (11 - 1)d = a + 10d = 5 ......(1)

13th term = a + (13 - 1)d = a + 12d = 79 ......(2)

from equations (1) and (2) we get,

2d = 79 - 5 = 74

⇒d = 37

(ii) a₁₀ - a₅ = 200

⇒{a + (10 -1)d} - {a + (5 - 1)d} = 200

⇒a + 9d - a - 4d = 200

⇒5d = 200

⇒d = 40

(iii) 20th term is 10 more than 18th term

i.e., 20th term = 10 + 18th term

⇒a +(20 - 1)d = 10 + a + (18 - 1)d

⇒a + 19d = 10 + a + 17d

⇒2d = 10

⇒d = 5