If 1 2 + 2 ⋅ 2 2 + 3 2 + 2 ⋅ 4 2 + 5 2 + 2 ⋅ 6 2 + ⋯ up to n terms equals 1575 , find n .

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MathematicsSequence & SeriesSpecial sequences/seriesMedium2 minQB

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If $1^{2} + 2 \cdot 2^{2} + 3^{2} + 2 \cdot 4^{2} + 5^{2} + 2 \cdot 6^{2} + \dots$ up to $n$ terms equals 1575 , find $n$ .

Options:

Answer:

14.00

Solution:

For even $n$ , odd-position terms contribute odd squares and even-position terms contribute twice even squares. Simplification gives $n (n + 1)^{2} /2 = 1575$ . Thus $n (n + 1)^{2} = 3150 = 14 \cdot 1 5^{2}$ , so $n = 14$ .

Stream:JEESubject:MathematicsTopic:Sequence & SeriesSubtopic:Special sequences/series

⏱ 2mℹ️ Source: QB

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