If the sum of the first 9 terms of an arithmetic sequence is 216 and its first term is 4,what is the common difference?

In an arithmetic sequence, the sum of the first \( n \) terms \( S_n \) is given by the formula:

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

Where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the number of terms.

We are given:
- \( S_9 = 216 \),
- \( a = 4 \),
- \( n = 9 \).

Substitute the values into the formula:

\[
216 = \frac{9}{2} \left( 2(4) + (9 - 1) d \right)
\]

Now, simplify and solve for \( d \):

\[
216 = \frac{9}{2} \left( 8 + 8d \right)
\]
\[
216 = \frac{9}{2} \times (8 + 8d)
\]
\[
216 = \frac{9}{2} \times 8(1 + d)
\]
  

If the sum of the first 9 terms of an arithmetic sequence is 216 and its first term is 4, what is the common difference?

Learn how to find the common difference of an arithmetic sequence where the sum of the first 9 terms is 216 and the first term is 4. This problem involves applying the arithmetic sequence sum formula and is suitable for students in Grades 9-12.

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