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Question

# If (8+3√7)n=P+F, where P is an integer and 0<F<1, then

A
P is an odd integer
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B
P is an even integer
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C
F.(P+F)=1
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D
(1F)(P+F)=1
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Solution

## The correct options are A (1−F)(P+F)=1 D P is an odd integerConsider the expansion of (8+3√7)n+(8−3√7)n. The alternate terms will cancel from both the expansions. Hence, (8+3√7)n+(8−3√7)n=2[8n+nC2(8)n−2(3√7)2+...] (The last term will depend on whether n is odd or even). Hence, (8+3√7)n+(8−3√7)n is even. We have, 0<8−3√7<1.Hence, 0<(8−3√7)n<1. (8+3√7)n=P+F, where 0<F<1 and P is an integer. Hence, P must be odd. Hence, (8−3√7)n=1−F.We have, (8+3√7)×(8−3√7)=1.Hence, (P+F)×(1−F)=1.Hence, options A and D are the correct answers.

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