To find the sum of the first 19 terms of an arithmetic sequence, we need the following information:

1. The 10th term: $a_{10} = 20$
2. The formula for the $n$-th term: $a_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference.
3. The sum of $n$ terms formula: $S_n = \frac{n}{2}(2a + (n-1)d)$.

Step 1: Express the 10th term in terms of $a$ and $d$:

$a_{10} = a + 9d = 20$

Step 2: Solve for $S_{19}$:

The sum of the first $n$ terms is given by: $S_n = \frac{n}{2}(2a + (n-1)d)$ Substituting $n = 19$: $S_{19} = \frac{19}{2}(2a + 18d)$

From the equation $a + 9d = 20$, express $a$ in terms of $d$: $a = 20 - 9d$

Substitute $a = 20 - 9d$ into $S_{19}$: $S_{19} = \frac{19}{2}(2(20 - 9d) + 18d)$ Simplify the expression: $S_{19} = \frac{19}{2}(40 - 18d + 18d)$ $S_{19} = \frac{19}{2}(40)$ $S_{19} = 19 \times 20 = 380$

Final Answer:

The sum of the first 19 terms is 380.

Would you like further explanation or an example with different parameters?

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5 Related Questions:

1. How do you find the first term and common difference if more terms are provided?
2. How would the solution change if the common difference was explicitly given?
3. Can you find the sum of an arithmetic series if the first and last terms are provided?
4. What is the formula for the sum of the first $n$ terms when $a$ and $d$ are unknown?
5. How can we graph an arithmetic sequence to visualize the progression?

Tip:

When working with arithmetic sequences, always isolate the first term and common difference early, as they determine all properties of the sequence!