Question : The sum of 10 terms of the arithmetic series is 390. If the third term of the series is 19, find the first term:
Option 1: 3
Option 2: 5
Option 3: 7
Option 4: 8
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Correct Answer: 3
Solution : Given: The sum of 10 terms of the arithmetic series is 390. So, $n=10$ The third term of the series is 19. Let the first term of the series be $a$. We know, sum of arithmetic progression (A.P.) = $\frac{n}{2}[2a+(n-1)d]$ $n^{th}$ term = $a+(n-1)d$ So, $3^{rd}$ term ⇒ $a+(3-1)d=19$ ⇒ $a+2d=19$ ----------------------------(1) Sum of 10 terms = $\frac{10}{2}[2a+(10-1)d] =390$ ⇒ $2a+9d=78$ ---------(2) Multiplying 9 with equation (1) and 2 with equation (2), we get, ⇒ $9a+18d=171$ ------------------------(3) ⇒ $4a+18d=156$ ------------------------(4) Subtracting (4) from (3), we get, $5a=15$ $\therefore a=3$ Hence, the correct answer is 3.
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