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

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}

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}