Given the following information about the arithmetic series:

• The first term, $a_1 = 12$
• The last term, $a_n = -78$
• The sum of the series, $S_n = -363$

Step 1: Formula for the sum of an arithmetic series

The sum of an arithmetic series is given by the formula: $S_n = \frac{n}{2} (a_1 + a_n)$ where:

• $S_n$ is the sum of the series,
• $n$ is the number of terms,
• $a_1$ is the first term, and
• $a_n$ is the last term.

We are given $S_n = -363$, $a_1 = 12$, and $a_n = -78$. Plugging these values into the sum formula: $-363 = \frac{n}{2} (12 + (-78))$ Simplifying: $-363 = \frac{n}{2} \times (-66)$ $-363 = -33n$ Solving for $n$: $n = \frac{-363}{-33} = 11$ So, the number of terms is $n = 11$.

Step 2: Finding the common difference

The formula for the $n$-th term of an arithmetic series is: $a_n = a_1 + (n - 1)d$ where $d$ is the common difference.

Substituting $a_n = -78$, $a_1 = 12$, and $n = 11$ into the formula: $-78 = 12 + (11 - 1)d$ $-78 = 12 + 10d$ Solving for $d$: $-78 - 12 = 10d$ $-90 = 10d$ $d = \frac{-90}{10} = -9$ Thus, the common difference is $d = -9$.

Final Answer:

• The number of terms in the series is $11$.
• The common difference is $-9$.

Would you like further details or have any questions? Here are five related questions you might find helpful:

1. How can we verify the sum of the series using the number of terms and common difference?
2. What happens to the sum if the common difference is positive instead of negative?
3. Can you derive the common difference if only the first term and sum are given?
4. How would the sum change if the number of terms increases while keeping the first term constant?
5. How do arithmetic series apply to real-world problems?

Tip: In arithmetic sequences, the common difference remains constant throughout, allowing easy predictions of future terms.