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  • Determine the AP whose fifth term is 19 and the difference of the eighth term from the thirteenth term is 20.

Determine the AP whose fifth term is 19 and the difference of the eighth term from the thirteenth term is 20.

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\text{Given } a_{5} = 19 \\ a + 4d = 19 \quad (\text{1}) \quad (\text{using } a_{n} = a + (n-1)d) \\ a_{13} - a_{8} = 20 \\ a + 12d - a - 7d = 20 \\ 5d = 20 \\ d = \frac{20}{5} \\ d = 4 \\ \text{Put } d = 4 \text{ in (1) we get} \\ a + 4(4) = 19 \\ a = 19 - 16 \\ a = 3 \\ a_{1} = a = 3 \\ a_{2} = a + d = 3 + 4 = 7 \\ a_{3} = a + 2d = 3 + 2(4) = 3 + 8 = 11 \\ a_{4} = a + 3d = 3 + 3(4) = 3 + 12 = 15 \\ \text{Hence the AP is: } 3, 7, 11, 15, \ldots

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