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Let a₁, a₂, a₃, be an arithmetic progression and S_k denote the sum of its first k terms. If a₃ = 14 and S₄ = 46 , then the value of _ k=1 ²⁴ a_ k+1 S_k + S_ k+1 is equal to _____ .

Correct answer

38

Step-by-step solution

Let the first term of the A.P. be a and the common difference be d . Given a₃ = 14 : a + 2d = 14 (1) Given S₄ = 46 : 4 2 (2a + 3d) = 46 2a + 3d = 23 (2) Multiplying equation (1) by 2 gives 2a + 4d = 28 . Subtracting equation (2) from this gives: (2a + 4d) - (2a + 3d) = 28 - 23 d = 5 Substituting d = 5 into (1): a + 10 = 14 a = 4 Now, consider the general term of the given series: a_ k+1 S_k + S_ k+1 We know that a_ k+1 = S_ k+1 - S_k . Substituting this into the numerator: S_ k+1 - S_k S_k + S_ k+1 Using the identi

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