The nth term of an arithmetic sequence t_n = t₁ + (n - 1)d, where t₁ is the first term of the arithmetic progression and 'd' is the common difference.

To find the 24th term of the arithmetic sequence, you will need the first term t₁ and the common difference 'd'.

Data given: 11th term and 31st term of the same arithmetic sequence.

1. Express the 11th term and the 31st term in terms of the first term and common difference of the arithmetic progression.
2. Solve the two equations to get the first term and the common difference.
3. Use the values obtained to find the 24th term of the arithmetic sequence

Step 1: Express the 11th and 31st term in terms of t₁ and d

or t₁₁ = 29 = t₁ + 10d ..... eqn (1)

The 31st term of the sequence t₃₁ = t₁ + (31 - 1)d

or t₃₁ = 169 = t₁ + 30d ...... eqn (2)

Subtract eqn (1) from eqn (2)

t₃₁ - t₁₁ = 169 - 29 =140 = t₁ + 30 d - (t₁ + 10d)

or 140 = 20d

So, d = 7.

Substitute the value of d in eqn (1)

29 = t₁ + 10*7

or t₁ = 29 - 70 = -41.

Step 2: Find the 24th term using t₁ and d computed in step 1

t₂₄ = t₁ + (24 - 1)d

t₂₄ = -41 + 23 * 7

t₂₄ = -41 + 161

  t₂₄ = 120

Alternatively: The difference between the 11th and 31st term is 20 common differences

The difference between any two consecutive terms of an arithmetic sequence is d. So, the difference between the 11th and the 31st term of an arithmetic sequence is 20 common differences.

t₃₁ – t₁₁ = 20d

169 – 29 = 140 = 20d

So, d= 7.

The difference between the 11th term and the 24th term is 13 common difference.

So, t₂₄ = t₁₁ + 13d

or t₂₄ = 29 + 13*7

Hence, t₂₄ = 29 + 91

  t₂₄ = 120