Question

The sum of the first twenty terms of an A.P. is 420. The ratio of the 5^th term to the 10^th term of A.P is 1 : 2. Find the 50^th term of the A.P.?
1. 100
2. 120
3. 140
4. 98

Answer 1

Correct Answer - Option 1 : 100

Given:

S₂₀ = 420

T₅ : T₁₀ = 1 : 2

Formula used:

T_n = a + (n – 1) d

S_n = n/2 × [2a + (n – 1) d]

Where, a = first term, n = n^th term of progression, d = common difference

Calculation:

S20 = n/2 × [2a + (n – 1) d]

⇒ 420 = 20/2 × [2a + 19d]

⇒ 2a + 19d = 42      ------ (1)

T₅ = a + (5 – 1) d

⇒ T₅ = a + 4d     ------ (2)

T₁₀ = a + (10- 1) d

⇒ T₁₀ = a + 9d      ------ (3)

Therefore,

T₅ : T₁₀ = 1 : 2

⇒ (a + 4d) : (a + 9d) = 1/2

⇒ 2a + 8d = a + 9d

⇒ a = d

Put this value in equation (1)

⇒ 2a + 19a = 42

⇒ 21a = 42

⇒ a = 2

∴ d = 2

So, 50^th term of the A.P., T₅₀ = a + (n – 1) d

⇒ T₅₀ = 2 + (50 – 1) 2

⇒ T₅₀ = 2 + 49 × 2

⇒ T₅₀ = 2 + 98

∴ T₅₀ = 100