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  • If sum of the 3rd and the 8th terms of an AP is 7 and the sum of the 7thand the 14thterms is –3, find the 10thterm.

If sum of the 3rd and the 8th terms of an AP is 7 and the sum of the 7thand the 14thterms is –3, find the 10thterm.

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It is given that the sum of 3\textsuperscript{rd} \text{ and } 8\textsuperscript{th} \text{ term is } 7.  

a_{3} = a + (3 - 1)d \quad (\text{using } a_{n} = a + (n-1)d) \\ a_{3} = a + 2d \\ a_{8} = a + (8 - 1)d \quad (\text{using } a_{n} = a + (n-1)d) \\ a_{8} = a + 7d \\ a_{3} + a_{8} = 7 \\ a + 2d + a + 7d = 7 \\ 2a + 9d = 7 \quad \ldots (1) \\ \text{Similarly, the sum of the } 7\textsuperscript{th} \text{ and } 14\textsuperscript{th} \text{ term is } -3 \\ a_{7} + a_{14} = -3 \\ a + (7 - 1)d + a + (14 - 1)d = -3 \quad (\text{using } a_{n} = a + (n-1)d) \\ a + 6d + a + 13d = -3 \\ 2a + 19d = -3 \quad \ldots (2)
  \\$Solve (1) and (2)$ 2a + 19d = - 3\\ 2a + 9d = 7\\ 10d = - 10\\ d = - 1\\ Put d = - 1 in (1)\\ \Rightarrow 2a + 9(-1) = 7 \\ \Rightarrow 2a = 7 + 9 \\ \Rightarrow a=\frac{16}{2}=8 \\ a\textsubscript{10} = (a + (10 - 1)d) \\ \Rightarrow a\textsubscript{10} = 8 + 9(-1) \\ \Rightarrow a\textsubscript{10} = 8 - 9\\ \Rightarrow a\textsubscript{10} = -1\\

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