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  • The eighth term of an AP is half its second term and the eleventh term exceeds one third of its fourth term by 1. Find the 15thterm.

The eighth term of an AP is half its second term and the eleventh term exceeds

one third of its fourth term by 1. Find the 15thterm.

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a_{8} = \frac{a_{2}}{2} \\ 2a_{8} = a_{2} \\ 2(a + 7d) = a + d \quad (\text{Using } a_{n} = a + (n - 1)d) \\ 2a + 14d - a - d = 0 \\ a + 13d = 0 \quad \ldots (1) \\ 3a_{11} = a_{4} + 3 \\ 3(a + 10d) = a + 3d + 3 \quad (\text{Using } a_{n} = a + (n - 1)d) \\ 3a + 30d - a - 3d = 3 \\ 2a + 27d = 3 \quad \ldots (2) \\ \text{Apply: eq.}(2) - 2 \times \text{eq.}(1) \text{ we get} \\ 2a + 27d - 2a - 26d = 3 \\ d = 3 \\ \text{Put } d = 3 \text{ in (1) we get} \\ a + 13(3) = 0 \\ a = -39 \\ \text{Then } a_{15} = a + 14d \quad (\text{Using } a_{n} = a + (n - 1)d) \\ = -39 + 14(3) \\ = -39 + 42 \\ = 3

 

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