The sum of the series below is 10,900 . How many numbers, , are in the series?

27
100
109
135

This series is an arithmetic progression where the first term and the last term . The common difference can be calculated as follows: .

To find the number of terms in the series, we can use the formula for the nth term of an arithmetic sequence:

Substituting the known values:

Rearranging gives:

Solving for :

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!