The first three terms of an arithmetic series add to 14 and the first 6 add to 21, what is the first term of the series.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The sum of the first n terms of an arithmetic series with first term a and common difference d is (n/2)*(2*a + (n - 1)*d)

Here, the sum of the first 3 terms is 14 and the sum of the first 6 terms is 21.

(3/2)*(2a + 2d) = 14 and 3*(2a + 5d) = 21

3*(2a + 5d) = 21

=> d = (7 - 2a)/5

Substitute in (3/2)*(2a + 2d) = 14

=> (3/2)*(2a + 2*(7 - 2a)/5) = 14

=> 2a + 14/5 - (4a)/5 = 28/3

=> (6a)/5 = 28/3 - 14/5

=> a = 49/9

The first term of the series is 49/9

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