Statement 1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + + (361 + 380 + 400) is 8000 .Statement 2: _k=1^n (k^3 - (k-1)^3) = - Mathematics MCQ AIEEE 2012 2026 | ExamTest.live

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Statement 1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + \dots + (361 + 380 + 400) is 8000 .Statement 2: \sum_{k = 1}^{n} (k^{3} - (k - 1)^{3}) = n^{3} for any natural number n .

A $Statement 1 is false, statement 2 is true$
B $Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1$
C $Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1$
D $Statement 1 is true, statement 2 is false$

Solution & Step-by-step Explanation

Statement 2: This is a telescoping sum: (1^{3} - 0^{3}) + (2^{3} - 1^{3}) + \dots + (n^{3} - (n - 1)^{3}) = n^{3} This is clearly true.Statement 1: Let's look at the general term T_{k} .The terms are: T_{1} = 1 = 1^{3} - 0^{3} T_{2} = 1 + 2 + 4 = 1^{2} + 1 \cdot 2 + 2^{2} = \frac{2 ^{3} - 1 ^{3}}{2 - 1} = 7 T_{3} = 4 + 6 + 9 = 2^{2} + 2 \cdot 3 + 3^{2} = \frac{3 ^{3} - 2 ^{3}}{3 - 2} = 19 The general term is T_{k} = (k - 1)^{2} + (k - 1) k + k^{2} .We know that k^{3} - (k - 1)^{3} = (k - (k - 1)) (k^{2} + k (k - 1) + (k - 1)^{2}) = T_{k} .The series ends at (361 + 380 + 400) = (1 9^{2} + 19 \cdot 20 + 2 0^{2}), which is T_{20} .Sum S = \sum_{k = 1}^{20} T_{k} = \sum_{k = 1}^{20} (k^{3} - (k - 1)^{3}) = 2 0^{3} = 8000 .Statement 1 is true, and Statement 2 correctly explains it.

Try it yourself before checking the explanation above.

Statement 1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + \dots + (361 + 380 + 400) is 8000 .Statement 2: \sum_{k = 1}^{n} (k^{3} - (k - 1)^{3}) = n^{3} for any natural number n .

Statement 1 is false, statement 2 is true

Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

Statement 1 is true, statement 2 is false

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