The mean of the first ten terms of an arithmetic sequence is 6. The mean of the first twenty terms of the arithmetic sequence is 16. Find the value of the 15th term of the sequence.

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justaguide eNotes educator| Certified Educator

The mean of the first ten terms of an arithmetic sequence is 6. The mean of the first twenty terms of the arithmetic sequence is 16.

For an arithmetic sequence with first term a and common difference d, the sum of the first n terms is `S_n = (n/2)*(2a + (n - 1)*d)`

Here `(S_10)/10 = 6 `

=> `5*(2a + 9*d) = 60`

=> 60 = 10a + 45d ...(1)

`(S_20)/20 = 16`

=> `10*(2a + 19*d) = 320`

=> 320 = 20a + 190d ...(2)

(2) - 2*(1)

=> 20a + 190d - 20a - 90d = 320 - 120

=> 100d = 200

=> d = 2

60 = 10a + 90

=> a = -3

The nth term of the series is equal to `T_n = a + (n - 1)*d`

`T_15 = -3 + 14*2 = 25`

The 15th term of the series is 25