Question

Find $t_{n}$ for the following A.P.s
i) 4, 9, 14, 19, ...
ii) $4,\frac{14}{3},\frac{16}{3},6,...$

Answer 1

Answer:

i) Tn = 5n - 1

ii) Tn = 2n/3 + 10/3.

Step-by-step explanation:

Hi,

i)

Given 4, 9, 14, 19,......

T₁ = 4

T₂ - T₁ = 9 - 4

= 5

T₃ - T₂ = 14 - 9

= 5

We can observe that all consecutive terms differ by

same common difference 'd' = 5

Tn = (Tn-1  - Tn-2) + (Tn-2 - Tn-3) +.........+(T₃ - T₂) + (T₂ - T₁) + T₁

We know all difference Tn-1  - Tn-2 = 5 and they are (n-1) times

Hence, Tn = (n - 1)*(T₂ - T₁) + T₁

= 4 + 5(n - 1)

Tn = 5n - 1.

i)

Given 4, 14/3, 16/3, 6,......

T₁ = 4

T₂ - T₁ = 14/3 - 4

= 2/3

T₃ - T₂ = 16/3 - 14/3

= 2/3

We can observe that all consecutive terms differ by

same common difference 'd' = 2/3

Tn = (Tn-1  - Tn-2) + (Tn-2 - Tn-3) +.........+(T₃ - T₂) + (T₂ - T₁) + T₁

We know all difference Tn-1  - Tn-2 = 5 and they are (n-1) times

Hence, Tn = (n - 1)*(T₂ - T₁) + T₁

= 4 + 2/3(n - 1)

Tn = 2n/3 + 10/3.

Hope, it helps !