We have the formula for the [tex]\( n \)[/tex]th term given by:
[tex]\[ a_n = \frac{1}{2n - 1} \][/tex]
We need to find the smallest integer [tex]\( n \)[/tex] for which [tex]\( a_n \)[/tex] is less than 0.01. This means we need to solve the inequality:
[tex]\[ \frac{1}{2n - 1} < 0.01 \][/tex]
To do this, we first find the condition on the denominator:
[tex]\[ 2n - 1 > \frac{1}{0.01} \][/tex]
Simplify [tex]\(\frac{1}{0.01}\)[/tex]:
[tex]\[ \frac{1}{0.01} = 100 \][/tex]
So, we have:
[tex]\[ 2n - 1 > 100 \][/tex]
Next, we solve for [tex]\( n \)[/tex]:
[tex]\[ 2n > 101 \][/tex]
[tex]\[ n > \frac{101}{2} \][/tex]
[tex]\[ n > 50.5 \][/tex]
Since [tex]\( n \)[/tex] must be an integer, we take the next whole number greater than 50.5, which is [tex]\( n = 51 \)[/tex].
To verify, we calculate the 51st term:
[tex]\[ a_{51} = \frac{1}{2 \cdot 51 - 1} \][/tex]
[tex]\[ a_{51} = \frac{1}{101} \][/tex]
[tex]\[ a_{51} = 0.009900990099009901 \][/tex]
Since 0.009900990099009901 is indeed less than 0.01, [tex]\( n = 51 \)[/tex] is the first term that meets the condition [tex]\( \frac{1}{2n - 1} < 0.01 \)[/tex].
Thus, the first term less than 0.01 is the 51st term.
