The first two terms of an arithmetic progression are 21 and 18.If the sum of the first $\displaystyle n$ th term is -24.Find the value of $\displaystyle n$
If you know the the first two terms, you know the initial term and the common difference. This Wikipedia article then gives you the formula for the sum of n terms, so you have to solve the equation for n.
The formula for a sum of a finite arithmetic series is:
$\displaystyle \frac{n(2a_1+d(n-1))}{2}$
In this formula, $\displaystyle a_1$ is the first term. n is the number of terms. d is the common difference (the rate at which the series progresses).
We know that the first term is 21, and we know that the common difference is -3, and that the sum of this series is -24.
Therefore, we have:
$\displaystyle \frac{n(42-3(n-1))}{2}=-24$
now we solve for n:
$\displaystyle n(42-3n+3)=-48$
$\displaystyle 42n-3n^2+3n=-48$
$\displaystyle 45n=3n^2-48$
divide by three to get:
$\displaystyle 15n=n^2-16$
$\displaystyle n^2-15n-16=0$
$\displaystyle (n-16)(n+1)=0$
Therefore:
$\displaystyle n-16=0$ or $\displaystyle n+1=0$
$\displaystyle n=16$ or $\displaystyle n=-1$
Since we cant have a negative number of terms, -1 is not a valid answer. Thus, we have :
$\displaystyle n=16$