Given:

1. The common difference $d = 5$.
2. The sum of the first 13 terms $S_{13} = 1326$.
3. The sum of the first 14 terms $S_{14} = 1463$.

We need to determine the first term $a$ of the arithmetic sequence.

Step 1: Recall the formula for the sum of the first $n$ terms of an arithmetic sequence:

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

For $n = 13$:

$S_{13} = \frac{13}{2} \left( 2a + 12d \right)$

Substituting the known values:

$1326 = \frac{13}{2} \left( 2a + 12 \times 5 \right)$

Simplifying:

$1326 = \frac{13}{2} \left( 2a + 60 \right)$

Multiply both sides by 2 to clear the fraction:

$2652 = 13 \left( 2a + 60 \right)$

Divide both sides by 13:

$204 = 2a + 60$

Subtract 60 from both sides:

$144 = 2a$

Divide by 2:

$a = 72$

Step 2: Verify with the Sum of the First 14 Terms

The formula for $S_{14}$ is:

$S_{14} = \frac{14}{2} \left( 2a + 13d \right)$

Substituting the known values:

$1463 = 7 \left( 2 \times 72 + 13 \times 5 \right)$

Calculate inside the parentheses:

$1463 = 7 \left( 144 + 65 \right)$

$1463 = 7 \times 209$

$1463 = 1463$

The calculations are correct, confirming that the first term $a = 72$.

Final Answer:

The first term $a$ is $\boxed{72}$.

Would you like more details or have any questions?

Related Questions:

1. How would the solution change if the common difference was negative?
2. What is the sum of the first 20 terms of this sequence?
3. How do you determine the number of terms required to reach a specific sum?
4. How would you solve this problem if only one sum was given?
5. How do you find the nth term of an arithmetic sequence given the first term and the common difference?

Tip:

In arithmetic sequences, using the sum formulas helps to cross-check results when working with sums of consecutive terms.