The sum of the series below is 10,900 . How many numbers, \( n \), are in the series? \( 19+20.5+22+23.5+\ldots+181 \) 27 100 109 135

This series is an arithmetic progression where the first term \( a = 19 \) and the last term \( l = 181 \). The common difference \( d \) can be calculated as follows: \( d = 20.5 - 19 = 1.5 \). To find the number of terms \( n \) in the series, we can use the formula for the nth term of an arithmetic sequence: \[ l = a + (n-1)d \] Substituting the known values: \[ 181 = 19 + (n-1)(1.5) \] Rearranging gives: \[ 162 = (n-1)(1.5) \] Solving for \( n \): \[ n-1 = \frac{162}{1.5} = 108 \quad \Rightarrow \quad n = 109 \] Thus, the number of terms in the series is 109. Now let’s delve into how this series can be applied in real-life scenarios! Arithmetic sequences, just like this one, can represent various phenomena, such as salary increments, where each period (like a year) a fixed amount is added to the initial salary. Cool, right? When dealing with arithmetic series, a common mistake people make is miscalculating the last term or common difference. Always double-check that you have the right values before plugging them into the formulas. It can save you a lot of time and potential mix-ups!