Identify the Given Sequence

The question asks for the average of natural numbers from $1$ to $92$. The sequence is $1, 2, 3, \dots, 92$. This is an Arithmetic Progression (AP) with first term $a = 1$, last term $l = 92$, and total terms $n = 92$.

$a = 1, \quad l = 92, \quad n = 92$

Recall the Sum Formula for First n Natural Numbers

The sum $S_n$ of the first $n$ natural numbers is given by the formula:

$S_n = \frac{n(n + 1)}{2}$

Derive the Average Formula

Average is calculated as the total sum divided by the number of terms ($n$). Therefore:

$\text{Average} = \frac{S_n}{n} = \frac{\frac{n(n + 1)}{2}}{n} = \frac{n + 1}{2}$

Substitute n = 92 into the Formula

Substitute $n = 92$ into the derived average formula $\frac{n + 1}{2}$:

$\text{Average} = \frac{92 + 1}{2} = \frac{93}{2} = 46.5$