1 what is S n for the Arithmetic series with d -4 a n 27 and n 9 2 What are the two means between 2 and 1 4?

For the arithmetic series with a common difference ( d = -4 ), first term ( a = 27 ), and ( n = 9 ), the sum ( S_n ) can be calculated using the formula ( S_n = \frac{n}{2} (2a + (n-1)d) ). Plugging in the values, we find ( S_9 = \frac{9}{2} (2 \times 27 + (9-1)(-4)) = \frac{9}{2} (54 - 32) = \frac{9}{2} \times 22 = 99 ).

The two means between 2 and 14 can be found by evenly spacing them in an arithmetic sequence. The common difference ( d ) can be calculated as ( d = \frac{b - a}{n + 1} = \frac{14 - 2}{3} = 4 ). Thus, the two means are ( 2 + 4 = 6 ) and ( 6 + 4 = 10 ), resulting in the means 6 and 10.