For a given arithmetic sequence, the 33rd term, a33, is equal to 285, and the 94th term, a94, is equal to 773. Find the value of the 12th term, a12.

Steps to Solve

1. Set Up Equations for Given Terms

We know the equations for the 33rd and 94th terms of the arithmetic sequence.

Using the formula ( a_n = a_1 + (n-1)d ):

• For the 33rd term: $$ a_{33} = a_1 + 32d $$ Which gives us: $$ a_1 + 32d = 285 \quad \text{(1)} $$

• For the 94th term: $$ a_{94} = a_1 + 93d $$ Which gives us: $$ a_1 + 93d = 773 \quad \text{(2)} $$

2. Subtract the Two Equations

To eliminate ( a_1 ), we can subtract the first equation from the second: $$ (a_1 + 93d) - (a_1 + 32d) = 773 - 285 $$

This simplifies to: $$ 61d = 488 $$

3. Solve for the Common Difference ( d )

Dividing both sides by 61: $$ d = \frac{488}{61} = 8 $$

4. Substitute ( d ) Back into One of the Equations

Now that we have ( d ), we can substitute it back into equation (1) to find ( a_1 ): $$ a_1 + 32(8) = 285 $$

This simplifies to: $$ a_1 + 256 = 285 $$

5. Solve for the First Term ( a_1 )

Subtracting 256 from both sides: $$ a_1 = 285 - 256 = 29 $$

6. Find the 12th Term ( a_{12} )

Using the formula for the 12th term: $$ a_{12} = a_1 + (12-1)d $$

Substituting the values: $$ a_{12} = 29 + 11(8) $$

This simplifies to: $$ a_{12} = 29 + 88 $$

7. Calculate the Final Value

Adding the values gives: $$ a_{12} = 117 $$