# Create an explicit formula to solve for the arithmetic series. Identify the first term and common...

## Question:

Create an explicit formula to solve for the arithmetic series. Identify the first term and common difference.

{eq}\sum_{m = 1}^{10}(-2 + \frac{4}{3}m) {/eq}

## Arithmetic Series:

• An arithmetic series is a series in which the terms of the series are sequenced in a particular pattern. Every two consecutive terms of the series have a constant difference and this difference is known as the common difference of the series.
• We can find the {eq}n^{th}{/eq} term of the series by subtractive the sum of {eq}n{/eq} terms from the sum of {eq}n+1{/eq} terms of the series.

We have the following given data

{eq}\hspace{1cm} S_m = \sum_{m = 1}^{10}(-2 + \frac{4}{3}m) {/eq}

• The first term of the series {eq}a_1=\,?{/eq}
• The common difference of the series {eq}d=\,?{/eq}

Solution

To find the first term, we need to put {eq}m=1{/eq} in the given summation formula.

\begin{align} \\ S_m & = \sum_{m = 1}^{10} \left(-2 + \frac{4}{3}m \right) \\[0.3cm] a_1 &=S_1\\[0.3cm] a_1 &= \sum_{m = 1}^{1} \left(-2 + \frac{4}{3}(1) \right) && \left[ \ \text{Put }~m=1 \ \right] \\[0.3cm] a_1 & = -\frac{2}{3} \\[0.3cm] \end{align}\\

Therefore, the first term of the series is {eq}\displaystyle \boxed{\color{blue} { \boldsymbol{ a_1 = -\frac{2}{3} } }} {/eq}

• The difference between any two consecutive terms of the series is known as the common difference.

\begin{align} \\ d &=S_{m+1}-S_m\\[0.3cm] d &=\left(-2 + \frac{4}{3}(m+1) \right) -\left(-2 + \frac{4}{3}m \right) \\[0.3cm] d &= -2 + \frac{4}{3}m+ \frac{4}{3} +2 - \frac{4}{3}m && \left[ \ \text{Simplifying }~ \ \right] \\[0.3cm] d &=\frac{4}{3}\\[0.3cm] \end{align}\\

Therefore, the common difference of the series is {eq}\displaystyle \boxed{\color{blue} { \boldsymbol{ d = \frac{4}{3} } }} {/eq}