The question is about an arithmetic sequence where:

• The 6th term ($a_6$) is 43.
• The 12th term ($a_{12}$) is 85.

We need to find the sum of the first 25 terms of the arithmetic sequence.

Step-by-Step Solution

1. General Formula for Arithmetic Sequence: The general form for the nth term in an arithmetic sequence is: $a_n = a_1 + (n - 1)d$ where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number.

2. Equations Based on Given Terms:

   • For the 6th term: $a_6 = a_1 + 5d = 43$
   • For the 12th term: $a_{12} = a_1 + 11d = 85$

3. Solving the System of Equations:

   We have two equations: $a_1 + 5d = 43 \quad \text{(Equation 1)}$ $a_1 + 11d = 85 \quad \text{(Equation 2)}$ Subtract Equation 1 from Equation 2: $(a_1 + 11d) - (a_1 + 5d) = 85 - 43$ $6d = 42 \quad \Rightarrow \quad d = 7$

4. Finding $a_1$: Substitute $d = 7$ into Equation 1: $a_1 + 5(7) = 43$ $a_1 + 35 = 43 \quad \Rightarrow \quad a_1 = 8$

5. Sum of the First 25 Terms:

   The sum of the first $n$ terms of an arithmetic sequence is given by: $S_n = \frac{n}{2} \times (2a_1 + (n - 1)d)$ Substituting $n = 25$, $a_1 = 8$, and $d = 7$: $S_{25} = \frac{25}{2} \times (2(8) + (25 - 1)(7))$ $S_{25} = \frac{25}{2} \times (16 + 24(7))$ $S_{25} = \frac{25}{2} \times (16 + 168)$ $S_{25} = \frac{25}{2} \times 184$ $S_{25} = 25 \times 92 = 2300$

Final Answer:

The sum of the first 25 terms is 2300.

Would you like further details on any step, or do you have additional questions?

Here are 5 related questions to expand on the concept:

1. How do you determine the common difference in any arithmetic sequence?
2. Can the sum formula be used for sequences with different numbers of terms?
3. How would the approach change if the sequence was geometric instead of arithmetic?
4. What if only one term and the common difference were given—how would you find the sum?
5. How can you calculate the sum if only the last term is known?

Tip: Always remember that the sum formula works only for arithmetic sequences. For other types of sequences, different formulas are needed!