Find the sum of the first 20 terms of the arithmetic series in which the 3rd term is 7 and 7th term is 2 more than three times its 3rd term.

# Find the sum of the first 20 terms of the arithmetic series in which the 3rd term is 7 and 7th term is 2 more than three times its 3rd term.

1. A
740
2. B
703
3. C
679
4. D
639

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### Solution:

Let a and d be the first term and the common difference of the AP.
${a}_{3}=a+2d=7$
${a}_{7}=a+6d=2+3{a}_{3}=2+3×7=23$
$⇒4d=23-7=16$
$⇒d=4$
$⇒a=-1$
${S}_{n}=\frac{n}{2}\left[2a+\left(n-1\right)d\right]=\frac{20}{2}\left[2×\left(-1\right)+19×4\right]=10×\left(76-2\right)=740$

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