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  • Verify that each of the following is an AP, and then write its next three terms. a, 2a + 1, 3a + 2, 4a + 3,...

Verify that each of the following is an AP, and then write its next three terms. a, 2a + 1, 3a + 2, 4a + 3,...

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\text{Here the given series is } a, 2a + 1, 3a + 2, 4a + 3, \ldots \\ a_{1} = a, \quad a_{2} = 2a + 1, \quad a_{3} = 3a + 2 \\ a_{2} - a_{1} = 2a + 1 - a = a + 1 \\ a_{3} - a_{2} = 3a + 2 - (2a + 1) = a + 1 \\ \text{Here, the common difference is the same; hence, the given series is an AP.} \\ a_{5} = a_{1} + 4d \quad (\text{using } a_{n} = a_{1} + (n-1)d) \\ = a + 4(a + 1) \\ = a + 4a + 4 \\ = 5a + 4 \\ a_{6} = a_{1} + 5d \\ = a + 5(a + 1) \\ = a + 5a + 5 \\ = 6a + 5 \\ a_{7} = a_{1} + 6d \\ = a + 6(a + 1) \\ = a + 6a + 6 \\ = 7a + 6 \\ \text{Next three terms are: } 5a + 4, 6a + 5, 7a + 6, \ldots

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