The sixth term of an arithmetic progression is 23 and the sum of the ﬁrst ten terms is 200. Find the seventh term.

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An arithmetic progression of initial term a and common difference d can be given as;

`T_n = a + (n-1)d` where n is the number of terms.

`T_6 = a+(6-1) d = 23`

`a+5d = 23` ----------------(1)

Sum of the arithmetic series described above is given by;

`S_n = (n/2) (2a+(n-1)d)`

`S_10 = (10/2) (2a+(10-1)d)=200`

`S_10= 5(2a+9d) = 200`

`2a+9d = 40` ----------------(2)

`(2) – (1) xx 2`

`-d = -6`

`d = 6`

From (1) `a = -7`

`T_7 = a + 6d = -7 +6xx6 = 31`

*So the seventh term of the series is 31.*

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