How do you find the 42nd term of an arithmetic sequence if the first term is 12 and the 27th term is 66?

To find the 42nd term of an arithmetic sequence where the first term a₁ = 12 and the 27th term a₂₇ = 66, we can follow these steps:

1. **Determine the common difference**:

In an arithmetic sequence, the nth term can be calculated with the formula:

a_n = a₁ + (n - 1) * d

Here, d is the common difference. Since we know the first term and the 27th term:

a₂₇ = a₁ + (27 - 1) * d

This gives us:

66 = 12 + 26d

To solve for d, subtract 12 from both sides:

66 - 12 = 26d

Which simplifies to:

54 = 26d

Now, divide both sides by 26:

d = 54 / 26 = 2.076923

2. **Calculate the 42nd term**:

Now that we have the common difference, we can use it to find the 42nd term:

a₄₂ = a₁ + (42 - 1) * d

Substituting in our values:

a₄₂ = 12 + 41 * 2.076923

Calculating this gives:

a₄₂ = 12 + 85.15479 = 97.15479

Thus, the 42nd term of the arithmetic sequence is approximately 97.15.