Question

# If X=8n−7n−1,n∈N and Y=49(n−1),n∈N, then -

A
X ⊆ Y
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B
Y ⊆ X
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C
X = Y
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D
None of these
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Solution

## The correct option is A X ⊆ YConventional Approach: Since 8n−7n−1=(7+1)n−7n−1 =7n+nC17n−1+nC27n−2+⋯+nCn−17+nCn−7n−1 =nC272+nC373+⋯+nCn7n,(nC0=nCn,nC1=nCn−1etc.) =49[nC2+nC3(7)+⋯+nCn7n−2] ∴ 8n−7n−1 is a multiple of 49 for n≥2 For n=1,8n−7n−1=8−7−1=0; For n=2,8n−7n−1=64−14−1=49 ∴8n−7n−1 is a multiple of 49 for all n∈N. ∴ X contains elements which are multiples of 49 and Y clearly contains all multiples of 49. ∴X⊂Y. Best Approach: Put n = 1, X = 0, Y = 0 For n = 2, X = 49, Y = 49 For n = 3, X = 490, Y = 98 ∴ X contains elements which are multiples of 49 and Y clearly contains all multiples of 49. ∴X⊂Y.

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