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- Arithmetic Sequences Calculator

This online tool can help you to 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.

Click here to see a list of example problems that can be solved by using this calculator.

**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 $\color{blue}{S_n = \frac{n}{2} \left(a_1 + a_n \right)}$.

**Problems that can be solved with this calculator **

**Example 1:**
Find the first term ($a_1$) of the arithmetic sequence having $a_8 = 5$ and $d = \frac{4}{3}$.

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**Example 2:**
The 12-th term of an arithmetic progression is 15 and the sum of the first 9 terms is 55.
Find the first term ($a_1$) and common difference ($d$).

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**Example 3:**
The first term of an arithmetic progression is -12, and the common difference is 3
determine how many terms must be added together to give a sum of 1104.

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