First slide

Special Series in Sequences and Series

Question

For $x \in R$ , let $[x]$ denote the greatest integer $\leq x .$ Largest natural number $n$ for which
$E = [\frac{π}{2}] + [\frac{1}{100} + \frac{π}{2}] + [\frac{2}{100} + \frac{π}{2}] \dots + [\frac{n}{100} + \frac{π}{2}] < 43,$ is

Moderate

A

41
B

42
C

43
D

97

Solution

Since $3.14 < π < 3.142, 1.57 < π / 2 < 1.571$
Thus, $[\frac{π}{2} + \frac{n}{100}] = 1$ for $n = 0, 1, 2, . . . . . ., 42$
Thus, largest natural number $n$ for which $E < 43$ is $41 .$

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