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  • Verify that each of the following is an AP, and then write its next three terms. a + b, (a + 1) + b, (a + 1) + (b + 1), ...

Verify that each of the following is an AP, and then write its next three terms. a + b, (a + 1) + b, (a + 1) + (b + 1), ...

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

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