Question

Write the expression a_n - a_k for the A.P. a, a + d, a + 2d, ..........

Hence, find the common difference of the A.P. for which

(i)11^th terms is 5 and 13^th term is 79.

(ii) a₁₀ - a₅ = 200

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

Answer 1

Let n^th term a_n = a + (n – 1) d

= a + nd – d

k^th term, a_k = a + (k – 1) d

= a + kd – d

Now,

a_n - a_k = (a + nd – d) – (a + kd – d)

= nd – kd = d (n – k)

(i) a₁₁ = 5

a + 10d = 5 (i)

a₁₃ = 79

a + 12d = 79 (ii)

By subtracting (i) from (ii), we get

2d = 74

d = 37

Hence, the common difference is 37

(ii) a₁₀ = a + 9d (i)

a₅ = a + 4d (ii)

a₁₀ – a₅ = a + 9d – a – 4d

200 = 5d

d = 40

Hence, common difference is 40

(iii) a₂₀ = a + 19d (i)

a₁₈ = a + 17d (ii)

Given that

a₂₀ = a₁₈ + 10

a + 19d = a + 17d + 10

2d = 10

d = 5

Hence, common difference is 5