We are given the following information about an arithmetic sequence:

• The sum of the first 10 terms is 225.
• The sum of the next 10 terms (from the 11th to the 20th term) is 525.

Let's break this down and find the first term of the sequence.

Step 1: Formula for the Sum of an Arithmetic Sequence

The sum $S_n$ of the first $n$ terms of an arithmetic sequence is given by the formula:

$S_n = \frac{n}{2} \left( 2a + (n-1)d \right)$ Where:

• $n$ is the number of terms.
• $a$ is the first term.
• $d$ is the common difference.

Step 2: Equation for the Sum of the First 10 Terms

For the first 10 terms, we have:

$S_{10} = \frac{10}{2} \left( 2a + (10-1)d \right) = 225$

Simplifying:

$5 \left( 2a + 9d \right) = 225$

$2a + 9d = 45 \quad \text{(Equation 1)}$

Step 3: Equation for the Sum of the Next 10 Terms

The sum of the next 10 terms (from the 11th to the 20th terms) is given by:

$S_{11-20} = \frac{10}{2} \left( 2(a + 10d) + (10-1)d \right) = 525$

Simplifying:

$5 \left( 2(a + 10d) + 9d \right) = 525$

Expanding and simplifying:

$5 \left( 2a + 20d + 9d \right) = 525$

$5 \left( 2a + 29d \right) = 525$

$2a + 29d = 105 \quad \text{(Equation 2)}$

Step 4: Solving the System of Equations

We now have two equations:

1. $2a + 9d = 45$
2. $2a + 29d = 105$

Subtract Equation 1 from Equation 2:

$(2a + 29d) - (2a + 9d) = 105 - 45$

Simplifying:

$20d = 60$

$d = 3$

Step 5: Finding the First Term

Substitute $d = 3$ into Equation 1:

$2a + 9(3) = 45$

$2a + 27 = 45$

$2a = 18$

$a = 9$

Final Answer:

The first term of the arithmetic sequence is $a = 9$.

Would you like more details or have any questions?

Here are 5 related questions to consider:

1. How would the sum change if the common difference was negative?
2. Can you derive the formula for the sum of an arithmetic sequence?
3. What would the sum of the first 15 terms be for this sequence?
4. How would the sequence look if the first term was $a = 12$?
5. What if the sum of the first 10 terms was 100 instead of 225?

Tip: When dealing with arithmetic sequences, the sum formula is especially useful in analyzing large sequences quickly!