Question

Find the 31st term of an A.P. whose 11th term is 38 and the 16th term is 73.

Answer 1

• The 31st term of an arithmetic progression (A.P.) = 178.

Given :

• The 11th term = 38.
• The 16th term = 73.

To Find :

• The 31st term.

Solution :

Given that,

• a₁₁ = 38.
• a₁₆ = 73.

We know that,

⇒ aₙ = a + (n - 1) d

First, we need to find the common difference (d),

Where,

• n = 11.

⇒ a₁₁ = a + (11 - 1) d

⇒ 38 = a - 10d ––––––––(1)

Similarly,

⇒ aₙ = a + (n - 1) d

Where,

• n = 16.

⇒ a₁₆ = a + (16 - 1) d

⇒ 73 = a + 15d ––––––––(2)

On subtracting (1) from (2).

We obtain,

⇒ 73 - 38 = a - 15d - a - 10d

⇒ 35 = 5d

⇒ 35 / 5 = d

⇒ 7 = d

Hence, the common difference (d) is 7.

Now, we need to find the value of a.

From equation (1),

⇒ 38 = a + 10d

⇒ 38 = a + 10 × 7

⇒ 38 = a + 70

⇒ 38 - 70 = a

⇒ -32 = a

Hence, the value of a is -32.

⇒ a₃₁ = a + (31 - 1) d

Where,

• a = -32.
• d = 7.

⇒ aₙ = a + (n - 1) d

Where,

• n = 31.
• a = -32.
• d = 7.

⇒ a₃₁ = -32 + 30 × 7

⇒ a₃₁ = -32 + 210

⇒ a₃₁ = 210 - 32

⇒ a₃₁ = 178

Hence,

The 31st term of an arithmetic progression (A.P.) is 178.