4. A student calculates the sum of the first 15 terms of an arithmetic sequence as 360. What is the value of the 20th term?

To find the value of the 20th term in the arithmetic sequence, let's break down the problem step by step:

### Step 1: Formula for the sum of an arithmetic sequence
The formula for the sum \( S_n \) of the first \( n \) terms of an arithmetic sequence is:

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

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

### Step 2: Using the given information
We are given:
- \( S_{15} = 360 \) (the sum of the first 15 terms is 360).
- \( n = 15 \) (the number of terms).

Substitute into the sum formula:

\[
360 = \frac{15}{2} \left( 2a + (15-1) d \right)
\]
\[
360 = \frac{15}{2} \left( 2a + 14d \right)
\]
Multiply both sides by 2 to eliminate the fraction:

\[
720 = 15 \left( 2a + 14d \right)
\]
Now, divide both sides by 15:

\[
48 = 2a + 14d
\]

This equation simplifies to:

\[
2a + 14d = 48 \tag{1}
\]

### Step 3: Find the 20th term
The general formula for the \( n \)-th term of an arithmetic sequence is:

\[
T_n = a + (n-1) d
\]

We are asked to find \( T_{20} \). So:

\[
T_{20} = a + (20-1) d = a + 19d
\]

### Step 4: Solve for \( a + 19d \)
From equation (1), we know that:

\[
2a + 14d = 48
\]

We can express \( 2a + 14d \) in terms of \( a + 19d \). Subtract equation (1) from \( a + 19d \) as follows:

\[
(a + 19d) = (a + 7d) + 12


A student calculates the sum of the first 15 terms of an arithmetic sequence as 360. What is the value of the 20th term?

Learn how to calculate the 20th term of an arithmetic sequence when given the sum of the first 15 terms as 360. This detailed solution involves using the arithmetic sequence sum formula and the nth term formula. Suitable for high school students studying algebra and arithmetic sequences.

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