# Math Help - Problem Involving Arithmetic progression

1. ## Problem Involving Arithmetic progression

The first two terms of an arithmetic progression are 21 and 18.If the sum of the first $n$ th term is -24.Find the value of $n$

2. 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.

3. The formula for a sum of a finite arithmetic series is:

$\frac{n(2a_1+d(n-1))}{2}$

In this formula, $a_1$ 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:

$\frac{n(42-3(n-1))}{2}=-24$

now we solve for n:

$n(42-3n+3)=-48$

$42n-3n^2+3n=-48$

$45n=3n^2-48$

divide by three to get:

$15n=n^2-16$

$n^2-15n-16=0$

$(n-16)(n+1)=0$

Therefore:

$n-16=0$ or $n+1=0$

$n=16$ or $n=-1$

Since we cant have a negative number of terms, -1 is not a valid answer. Thus, we have :

$n=16$

4. Originally Posted by mastermin346
The first two terms of an arithmetic progression are 21 and 18.

If the sum of the first $n$ th term is -24, find the value of $n$
Can you just confirm that that is the "sum of the first n terms"
rather than the "sum of the first and nth terms" ?