Arithmetic Progressions - Introduction

For the AP: $5, 2, -1, -4, \dots$, find the first term $a$ and the common difference $d$.

The first term $a = 5$. The common difference $d$ is calculated as $a_2 - a_1 = 2 - 5 = -3$. Therefore, $a = 5$ and $d = -3$.

In an AP, the first value listed is $a$, and $d$ is found by subtracting any term from the term that follows it.

Find the $12^{th}$ term of the AP where $a = 10$ and $d = 7$.

Using the formula $a_n = a + (n - 1)d$: $a_{12} = 10 + (12 - 1)7$ $a_{12} = 10 + (11 \times 7)$ $a_{12} = 10 + 77$ $a_{12} = 87$.

Substitute the values of the first term $a$, the position $n$, and the common difference $d$ into the general term formula.

Check if $301$ is a term of the list of numbers $5, 11, 17, 23, \dots$.

Here $a = 5$ and $d = 11 - 5 = 6$. Let $a_n = 301$. $301 = 5 + (n - 1)6$ $296 = (n - 1)6$ $n - 1 = \frac{296}{6} = \frac{148}{3}$ $n = \frac{148}{3} + 1 = \frac{151}{3}$. Since $n$ must be a positive integer and $\frac{151}{3}$ is not an integer, $301$ is not a term of the AP.

The position of a term $n$ must always be a natural number. If solving for $n$ results in a fraction, that value does not exist in the sequence.

Calculate the difference between the $50^{th}$ term and $30^{th}$ term of an AP with $d = 5$.

$a_{50} - a_{30} = [a + (50-1)d] - [a + (30-1)d]$ $= (a + 49d) - (a + 29d)$ $= 20d$ Since $d = 5$: $20 \times 5 = 100$.

The difference between any two terms $a_n$ and $a_m$ is always $(n-m)d$.