Answer – The explicit formula for the arithmetic sequence $\mathbf{\{}\mathbf{5}\mathbf{,}\mathbf{ }\mathbf{10}\mathbf{,}\mathbf{ }\mathbf{15}\mathbf{,}\mathbf{ }\mathbf{20}\mathbf{,}\mathbf{ }\mathbf{\dots }\mathbf{\}}$ can be found using the base formula ${\mathit{a}}_{\mathbf{n}}\mathbf{ }\mathbf{=}\mathbf{ }\mathit{c}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{(}\mathit{n}\mathbf{ }\mathbf{–}\mathbf{ }\mathbf{1}\mathbf{)}\mathbf{ }\mathit{d}$ as ${\mathit{a}}_{\mathbf{n}}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{5}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{(}\mathit{n}\mathbf{ }\mathbf{–}\mathbf{1}\mathbf{)}\mathbf{ }\mathbf{5}$.

Explanation:

An arithmetic sequence is an ordered list of numbers in which any two consecutive numbers have the same difference between them. Each number in the sequence is referred to as a term, and the common difference between them is denoted by the letter d.

The explicit formula of such a sequence can be used to find any term (referred to as the nᵗʰ term, where n indicates the location of the term) in the sequence.

Let us consider an arithmetic sequence where the first term is a₁.

So according to the definition, since the common difference between each term is d, the second term $a_2 = a_1 + d$.

Similarly, the third term $a_3 = a_1 + d + d = a_1 + 2d$

The fourth term $a_4 = a_1 + d + d + d = a_1 + 3d$ and so on.

From this, we understand that for every term aₙ, we add d one time less than the index n of the term, i.e., (n – 1). For example, in the case of a₄ (with index 4), we add d thrice, which is 4 – 1.

By this logic, the nᵗʰ term in the sequence can be denoted by $a_n = a_1 + (n – 1) d$. We can also alternatively write this as $a_n = c + (n – 1) d$, where c is a₁.

This is the base formula we can use to arrive at the explicit formula for any arithmetic sequence.

Now, let’s come back to the given arithmetic sequence $\{5, 10, 15, 20, \dots \}$. Here, the first term $c = 5$ and the common difference $d = 5$.

We can now apply the base formula for an arithmetic sequence derived earlier to get the explicit formula for the given sequence:

$a_n = c + (n – 1) d$

$a_n = 5 + (n – 1) 5$

We can also verify the formula by substituting the first few values of n:

$a_1 = 5 + (1 – 1) 5 = 5$

$a_2 = 5 + (2 – 1) 5 = 5 + 5 = 10$

$a_3 = 5 + (3 – 1) 5 = 5 + 10 = 15$

So the formula for the given arithmetic sequence is $a_n = 5 + (n – 1) 5$.

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