The sum of 10 terms of the arithmetic series is 390. If the third term of the series is 19, find the first term.
3
To find the first term of the arithmetic series, let's start by using the formula for the sum of an arithmetic series:
\( S_n = \frac{n}{2} \times (2a + (n - 1)d) \)
Given:
- Sum of 10 terms \( S_{10} = 390 \)
- Third term \( a + 2d = 19 \)
Step 1: Write the sum formula for 10 terms:
\( 390 = \frac{10}{2} \times (2a + 9d) \)
Simplifying:
\( 390 = 5 \times (2a + 9d) \)
Divide both sides by 5:
\( 78 = 2a + 9d \) ... (Equation 1)
Step 2: Use the equation for the third term:
\( a + 2d = 19 \) ... (Equation 2)
Step 3: Solve the system of equations:
From Equation 2, express \( a \):
\( a = 19 - 2d \)
Substitute \( a \) in Equation 1:
\( 78 = 2(19 - 2d) + 9d \)
Distribute and simplify:
\( 78 = 38 - 4d + 9d \)
Combine like terms:
\( 78 = 38 + 5d \)
Subtract 38 from both sides:
\( 40 = 5d \)
Divide both sides by 5:
\( d = 8 \)
Step 4: Find \( a \) using the value of \( d \):
\( a = 19 - 2 \times 8 \)
\( a = 19 - 16 \)
\( a = 3 \)
Thus, the first term of the series is 3.
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