Write the expression an - ak for the A.P. a, a + d, a + 2d, ..........

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

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»

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