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Arithmetic sequences calculator
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
$$ a_{ 12 } = 15 ~~,~~ S_{ 9 } = 55 ~~,~~ a_1 = ? ~~,~~ d = ? $$solution
$$ a_1 = \frac{ 65 }{ 63 } ~~,~~ d = \frac{ 80 }{ 63 } $$explanation
After substituting $ n = 12 $ into $ a_n = a_1 + (n-1)d $ we have $ a_{ 12 } = a_1 + 11 d $, so $ a_1 = a_{ 12 } - 11 d = 15 - 11 d $.
After substituting this result into $ S_n = \frac{n}{2} ( 2a_1 + (n-1)d) $ we have,
$$ \begin{aligned} S_{ n } &= \frac{ 9 }{2} ( 2 \color{blue}{a_1} + ( n-1)d) \\[1 em] S_{ 9 } &= \frac{ 9 }{2} \left( 2 ( \color{blue}{ 15 - 11 d } ) + ( 9-1) d \right) \\[1 em] 55 &= \frac{ 9 }{2} \cdot \left( 30 - 22 \cdot d + 8 d \right) \\[1 em] 55 &= \frac{ 9 }{2} \cdot \left( 30 -14 d \right) \\[1 em] \frac{2\cdot 55 }{ 9 } &= 30 -14 d \\[1 em] \frac{ 110 }{ 9 } &= 30 -14 d \\[1 em] -14 d &= -\frac{ 160 }{ 9 } \\[1 em] d &= \frac{ 80 }{ 63 } \end{aligned}$$Now we will find $ a_1 $:
$$ a_1 = 15 - 11 d = 15 - 11 \cdot \frac{ 80 }{ 63 } = \frac{ 65 }{ 63 }$$About 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 $\color{blue}{S_n = \frac{n}{2} \left(a_1 + a_n \right)}$.
Please tell me how can I make this better.