Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Nov 07, 2023

Find the number of elements in the set $A = \{1, 3, 5, \ldots, 31\}$ .

STEP 1

Assumptions1. The set A contains all odd numbers from1 to31 inclusive. . The cardinal number of a set is the number of elements in the set.

STEP 2

The set A is an arithmetic sequence, where the common difference is2 (each number is2 greater than the previous one).We can use the formula for the nth term of an arithmetic sequence to find the number of terms. The formula is $a_n = a1 + (n -1) * d$ where- $a_n$ is the nth term, - $a1$ is the first term, - $d$ is the common difference, - and n is the number of terms.

STEP 3

In this case, we know that $a1 =1$ , $d =2$ , and $a_n =31$ . We can plug these values into the formula and solve for n.
$31 =1 + (n -1) *2$

STEP 4

implify the equation and solve for n.
$31 -1 = (n -1) 2$ $30 = (n -1) 2$ $n -1 =30 /2$ $n -1 =15$ $n =15 +1$ $n =16$ The cardinal number of the set A is16.

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