Find the next number in the series. 5,6,11,34,135, ? (a) 679...

Find the next number in the series. $$ 5,6,11,34,135, ? $$ (a) 679 (b) 677 (c) 678 (d) 676 (a) 16 (b) 8 (c) 4 (d) 32 38. If 2 is added to the numerator of a fraction, it reduces to $\left(\frac{1}{2}\right)$ and if 1 is subtracted from the denominator; it reduces to $\left(\frac{1}{3}\right)$. Find the fraction. (a) $\frac{5}{7}$ (b) $\frac{1}{10}$ (c) $\frac{2}{5}$ (d) $\frac{3}{10}$ 39. If 0.090909 is approximately equal to $\frac{1}{11}$, then the approximate value of 0.454545 is (a) $\frac{4}{11}$ (b) $\frac{5}{11}$ (c) $\frac{45}{11}$ (d) $\frac{6}{11}$ 40. Find the solution of the inequality $37-(3 x+5) \geq 9 x-8(x-3)$ (a) $[2,8]$ (b) $(-\infty, 2]$ (c) $(-\infty, 8]$ (d) $(-\infty, 8)$ 41. Find the solution of the inequality $\frac{1}{|x|-3} \leq \frac{1}{2}$ (a) $(-\infty,-6] \cup(-2,2) \cup[6, \infty)$ (b) $(-\infty,-5] \cup(-3,3) \cup[5, \infty)$ (c) $(-\infty,-6] \cup(-4,4)$ (d) $(-\infty,-6] \cup(-4,-4)$

Find the next number in the series.
$$
5,6,11,34,135, ?
$$
(a) 679
(b) 677
(c) 678
(d) 676

(a) 16
(b) 8
(c) 4
(d) 32
38. If 2 is added to the numerator of a fraction, it reduces to $\left(\frac{1}{2}\right)$ and if 1 is subtracted from the denominator; it reduces to $\left(\frac{1}{3}\right)$. Find the fraction.
(a) $\frac{5}{7}$
(b) $\frac{1}{10}$
(c) $\frac{2}{5}$
(d) $\frac{3}{10}$
39. If 0.090909 is approximately equal to $\frac{1}{11}$, then the approximate value of 0.454545 is
(a) $\frac{4}{11}$
(b) $\frac{5}{11}$
(c) $\frac{45}{11}$
(d) $\frac{6}{11}$
40. Find the solution of the inequality $37-(3 x+5) \geq 9 x-8(x-3)$
(a) $[2,8]$
(b) $(-\infty, 2]$
(c) $(-\infty, 8]$
(d) $(-\infty, 8)$
41. Find the solution of the inequality $\frac{1}{|x|-3} \leq \frac{1}{2}$
(a) $(-\infty,-6] \cup(-2,2) \cup[6, \infty)$
(b) $(-\infty,-5] \cup(-3,3) \cup[5, \infty)$
(c) $(-\infty,-6] \cup(-4,4)$
(d) $(-\infty,-6] \cup(-4,-4)$

Key Concepts

Sequence Pattern Recognition

This concept involves identifying patterns in sequences by examining the process that generates each term from the previous ones. It includes analyzing differences, ratios, or applying recursive rules to determine the rule behind the sequence.

Algebraic Manipulation of Fractions

This involves altering the numerators and denominators as per given instructions, and then setting up equations to solve for unknown values in fractions. It is crucial for problems where modifications to a fraction yield new conditions.

Conversion of Repeating Decimals to Fractions

This concept focuses on transforming repeating decimals into their fractional form by leveraging the properties of geometric series or setting up algebraic equations. This is essential for understanding the exact value of periodic decimal representations.

Solving Linear Inequalities

This key area involves isolating the variable in inequalities and determining the solution set. The process requires careful manipulation of the inequality signs when performing arithmetic operations, and is foundational in algebra.

Absolute Value Inequalities

This topic covers the challenges of handling inequalities that include absolute value expressions. It typically requires splitting the inequalities into separate cases based on the definition of absolute value, and establishing domain restrictions to correctly determine the solution set.

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