Binomial TheoremHard
Question
The greatest integer less than or equal to (√2 + 1)6 is
Options
A.196
B.197
C.198
D.199
Solution
T22 is the numerically greatest term.
(√2 + 1)6 = I + f
(√2 - 1)6 = f′
2[6C0 + 6C2 . 2 + 6C4 (2)2 + ........] = I + f + f′
f + f′ = 1 or f′ = 1 - f
I = 2 [6C0 + 6C2 .2 + 6C4 .4 + 6C6.8] - 1
I = 2 [1 + 30 + 60 + 8] - 1 = 197
(√2 + 1)6 = I + f
(√2 - 1)6 = f′
2[6C0 + 6C2 . 2 + 6C4 (2)2 + ........] = I + f + f′
f + f′ = 1 or f′ = 1 - f
I = 2 [6C0 + 6C2 .2 + 6C4 .4 + 6C6.8] - 1
I = 2 [1 + 30 + 60 + 8] - 1 = 197
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