Concept of Arithmetic Progression

Numerical sequence — an ordered set of numbers, in which a rule for generating all elements is specified.

Examples of different numerical sequences:

• \(1, 2, 3, 4, 5, \ldots\) — sequence of natural numbers;
• \(1, 1, 2, 3, 5, 8, \ldots\) — Fibonacci sequence, where each element from the third onward is the sum of the two previous elements;
• \(1, 4, 1, 5, 9, \ldots\) — digits after the decimal point of \( \pi \), with no obvious pattern.

Task: Find the fifteenth element in each of the sequences listed above.

Arithmetic progression — a sequence in which each next element differs from the previous one by the same number. Elements of a sequence are usually denoted by lowercase Latin letters with subscripts indicating the position of the number in the list. For example, in the progression \[1, 4, 7, 10, 13, \ldots\] the notation \(a_3\) refers to the third number in this list, that is \(a_3 = 7\). The number by which consecutive elements of a progression differ is called its common difference and is denoted by d. In our example, \(d = 3\).
Task: Find the fifteenth element of this progression.

The general term of an arithmetic progression can be described by the following formula:
\( a_n = a_1 + (n - 1)d \)

Task: Use this formula to find the fifteenth element of the progression above.