Write the expression an – ak for the A.P. a, a+d, a+2d, …..Hence

1.	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 term is 5 and 13th term is 79.(ii) a10 – a5 = 200(iii) 20th term is 10 more than the 18th term.
Answer» Given A.P. a, a+d, a+2d, ….. So, a_n= a + (n-1)d = a + nd –d And, a_k= a + (k-1)d = a + kd – d a_n– a_k= (a + nd – d) – (a + kd – d) = (n – k)d (i) Given 11^th term is 5 and 13^th term is 79, Here n = 13 and k = 11, a₁₃– a₁₁= (13 – 11)d = 2d ⇒ 79 – 5 = 2d d = 74/2 = 37 (ii) Given, a₁₀– a₅ = 200 ⇒ (10 – 5)d = 200 5d = 200 d = 40 (iii) Given, 20^th term is 10 more than the 18^th term. ⇒ a₂₀ – a₁₈ = 10 (20 – 18)d = 10 2d = 10 d = 5

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 term is 5 and 13th term is 79.(ii) a10 – a5 = 200(iii) 20th term is 10 more than the 18th term.

Answer»

Given A.P. a, a+d, a+2d, …..

So, a_n= a + (n-1)d = a + nd –d

And, a_k= a + (k-1)d = a + kd – d

a_n– a_k= (a + nd – d) – (a + kd – d)

= (n – k)d

(i) Given 11^th term is 5 and 13^th term is 79,

Here n = 13 and k = 11,

a₁₃– a₁₁= (13 – 11)d = 2d

⇒ 79 – 5 = 2d

d = 74/2 = 37

(ii) Given, a₁₀– a₅ = 200

⇒ (10 – 5)d = 200

5d = 200

d = 40

(iii) Given, 20^th term is 10 more than the 18^th term.

⇒ a₂₀ – a₁₈ = 10

(20 – 18)d = 10

2d = 10

d = 5