Advanced Sum Applications

When we know the sum ($S_n$), the first term ($a$), and the common difference ($d$) of an Arithmetic Progression, finding the number of terms ($n$) requires a bit of algebra.

Substituting known values into $S_n = \frac{n}{2}[2a + (n-1)d]$ often creates a quadratic equation in the form of $An^2 + Bn + C = 0$. Solving this equation gives us the possible values for $n$.

Problem

How many terms of the AP 24, 21, 18, $\dots$ must be taken so that their sum is 78?

Problem

A manufacturer of TV sets produced 600 sets in the third year and 700 sets in the seventh year. Assuming the production increases uniformly by a fixed number every year, find: (i) the production in the 1st year (ii) the production in the 10th year (iii) the total production in the first 7 years