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
Find $ a_1 $ (first term of arithmetic progression) if $ d = \frac{ 4 }{ 3 } ~~ \text{and} ~~ a_{ 8 } = 5 $.
solution
$$ a_1 = -\frac{ 13 }{ 3 } $$explanation
To find $ a_1 $ we use formula
$$ \color{blue}{a_n = a_1 + (n-1)d}$$For $ n = 8 $ the formula is:
$$ a_{ 8 } = a_1 + (8 - 1)d $$In this example we have $ d = \frac{ 4 }{ 3 } $ and $ a_{ 8 } = 5 $. After substituting these values to above formula, we obtain:
$$ \begin{aligned} a_{ 8 } &= a_1 + (8 - 1)d \\[1 em] 5 &= a_1 + 7 \cdot \frac{ 4 }{ 3 } \\[1 em] 5 &= a_1 + \frac{ 28 }{ 3 } \\[1 em] a_1 &= -\frac{ 13 }{ 3 } \end{aligned} $$The first few terms of this sequence are:
$$ -\frac{ 13 }{ 3 }, ~~~-3, ~~~-\frac{ 5 }{ 3 }, ~~~-\frac{ 1 }{ 3 }, ~~~1 . . . $$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 $\color{blue}{S_n = \frac{n}{2} \left(a_1 + a_n \right)}$.
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