This online tool can help you find $n^{th}$ term and the sum of the first $n$ terms of an arithmetic progression. Also, this calculator can be used to solve much more complicated problems. For example, the calculator can find the common difference ($d$) if $a_5 = 19 $ and $S_7 = 105$. The biggest advantage of this calculator is that it will generate all the work with detailed explanation.
problem
$$ a_1 = -12 ~,~ d = 3 ~,~ S_n = 1104 ~,~ n = ? $$solution
$$ ~~n = 32 $$explanation
To find $ n $ we use formula
$$ \color{blue}{S_n = \frac{n}{2} (2 a_1 + (n-1)d) }$$In this example we have $ a_1 = -12 ~~,~~ d = 3 ~~,~~ S_n = 1104 $. After substituting these values to above formula, we obtain:
$$ \begin{aligned} S_n &= \frac{n}{2} (2 a_1 + (n-1)d) \\ 2 \cdot -12 &= n (2 \cdot \left( -12 \right) + (n-1) \cdot 3) \\ 2208 &= n (3 \cdot n -27) \\ 3n^2-27n-2208 &= 0 \end{aligned}$$The solutions of this quadratic equation are $ n_1 = -23 $ and $ n_2 = 32 $ (click here to see detailed explanation on how to slove quadratic equation).
The first few terms of this sequence are:
$$ -12, ~~~-9, ~~~-6, ~~~-3, ~~~0 . . . $$Definition:
Arithmetic sequence is a list of numbers where each number is equal to the previous number, plus a constant. The constant is called the common difference ($d$).
Formulas:
The formula for finding $n^{th}$ term of an arithmetic progression is $\color{blue}{a_n = a_1 + (n-1) d}$, where $\color{blue}{a_1}$ is the first term and $\color{blue}{d}$ is the common difference.
The formulas for the sum of first $n$ numbers are $\color{blue}{S_n = \frac{n}{2} \left( 2a_1 + (n-1)d \right)}$ and $\color{blue}{S_n = \frac{n}{2} \left(a_1 + a_n \right)}$.
Please tell me how can I make this better.