We are given the following information about an arithmetic sequence:

• The sum of the first $n$ terms, $S_n = 438$,
• The last term, $a_n = 75$,
• The common difference, $d = 7$.

We need to find the first term $a_1$ and the number of terms $n$.

Step 1: Formula for the sum of an arithmetic sequence

The formula for the sum of the first $n$ terms of an arithmetic sequence is:

$S_n = \frac{n}{2} \times (a_1 + a_n)$

Substituting the given values:

$438 = \frac{n}{2} \times (a_1 + 75)$

Step 2: Expressing $a_n$ in terms of $a_1$ and $n$

The formula for the $n$-th term of an arithmetic sequence is:

$a_n = a_1 + (n - 1) \times d$

Substituting $a_n = 75$ and $d = 7$:

$75 = a_1 + (n - 1) \times 7$

Simplifying:

$75 = a_1 + 7n - 7$

$75 = a_1 + 7n - 7$

$a_1 = 82 - 7n$

Step 3: Substituting into the sum formula

Now, substitute $a_1 = 82 - 7n$ into the sum formula:

$438 = \frac{n}{2} \times ((82 - 7n) + 75)$

Simplifying:

$438 = \frac{n}{2} \times (157 - 7n)$

$438 = \frac{n(157 - 7n)}{2}$

Multiplying both sides by 2 to eliminate the fraction:

$876 = n(157 - 7n)$

Expanding:

$876 = 157n - 7n^2$

Rearranging to form a quadratic equation:

$7n^2 - 157n + 876 = 0$

Step 4: Solving the quadratic equation

We can now solve this quadratic equation using the quadratic formula:

$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $7n^2 - 157n + 876 = 0$, the coefficients are:

• $a = 7$,
• $b = -157$,
• $c = 876$.

Substituting these values into the quadratic formula:

$n = \frac{-(-157) \pm \sqrt{(-157)^2 - 4 \times 7 \times 876}}{2 \times 7}$

$n = \frac{157 \pm \sqrt{24649 - 24528}}{14}$

$n = \frac{157 \pm \sqrt{121}}{14}$

$n = \frac{157 \pm 11}{14}$

Thus, we have two possible values for $n$:

$n = \frac{157 + 11}{14} = \frac{168}{14} = 12$

$n = \frac{157 - 11}{14} = \frac{146}{14} = 10.43$

Since $n$ must be a whole number, we take $n = 12$.

Step 5: Finding $a_1$

Now that we know $n = 12$, substitute this value into the expression for $a_1$:

$a_1 = 82 - 7n$

$a_1 = 82 - 7 \times 12 = 82 - 84 = -2$

Final Answer:

The first term is $a_1 = -2$ and the number of terms is $n = 12$.

Would you like further details or have any questions?

Additional Practice:

1. Find the sum of the first 20 terms of an arithmetic sequence with $a_1 = 5$ and $d = 3$.
2. If the sum of the first 15 terms of an arithmetic sequence is 300 and the common difference is 4, find $a_1$ and $a_{15}$.
3. In an arithmetic sequence, the 5th term is 12, and the 10th term is 32. Find the common difference and $a_1$.
4. Find the number of terms in an arithmetic sequence where $a_1 = 8$, $d = 2$, and the sum is 210.
5. If the sum of an arithmetic sequence is 500 and the common difference is 5, what is the value of $n$ and $a_1$?

Tip: When solving quadratic equations in arithmetic problems, always verify that your final answer for $n$ is a whole number.