The sum of 10 terms of the arithmetic series is 390. If the third term of the series is 19, find the first term

Options

A. 3
B. 5
C. 7
D. 8

Answer

To solve the problem step by step, we will use the properties of arithmetic series.

**Step 1: Understand the given information.**
We know:
- The sum of the first 10 terms (S₁₀) = 390
- The third term (T₃) = 19

**Step 2: Write the formulas for the sum of an arithmetic series and the nth term.**
The sum of the first n terms of an arithmetic series can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (2A + (n - 1)D) \]
Where:
- \( S_n \) = sum of the first n terms
- \( A \) = first term
- \( D \) = common difference
- \( n \) = number of terms

The nth term of an arithmetic series is given by:
\[ T_n = A + (n - 1)D \]

**Step 3: Set up the equations based on the information provided.**
For the sum of the first 10 terms:
\[ S_{10} = \frac{10}{2} \times (2A + 9D) = 390 \]
This simplifies to:
\[ 5(2A + 9D) = 390 \]
Dividing both sides by 5 gives:
\[ 2A + 9D = 78 \] (Equation 1)

For the third term:
\[ T_3 = A + 2D = 19 \] (Equation 2)

**Step 4: Solve the equations simultaneously.**
From Equation 2, we can express \( A \) in terms of \( D \):
\[ A = 19 - 2D \]

Now, substitute \( A \) in Equation 1:
\[ 2(19 - 2D) + 9D = 78 \]
This simplifies to:
\[ 38 - 4D + 9D = 78 \]
Combining like terms gives:
\[ 38 + 5D = 78 \]
Subtracting 38 from both sides results in:
\[ 5D = 40 \]
Dividing by 5 gives:
\[ D = 8 \]

**Step 5: Substitute back to find the first term \( A \).**
Now that we have \( D \), substitute it back into the equation for \( A \):
\[ A = 19 - 2(8) \]
\[ A = 19 - 16 \]
\[ A = 3 \]

**Final Answer:**
The first term \( A \) of the arithmetic series is 3.

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