The mean and range of ten consecutive terms in an arithmetic progression are all 82. What is the first term in the sequence?

mathtoo
Aug 3, 2018

#1**0 **

**The mean and range of ten consecutive terms in an arithmetic progression are all 82. What is the first term in the sequence?**

\(\text{Let $a$ is the first term} \\ \text{Let $b$ is the last term}\)

\(\begin{array}{|lrcll|} \hline \text{mean:} & \dfrac{a+b}{2} &=& 82 \quad & | \quad \cdot 2 \\\\ & a+b &=& 2\cdot 82 \qquad (1) \\ \\ \text{range:} & b-a &=& 82 \qquad (2) \\ \hline \\ (1)-(2):& (a+b) - (b-a) &=& 2\cdot 82 - 82 \\ & a+b - b+a &=& 82 \\ & a+a &=& 82 \\ & 2a &=& 82 \quad & | \quad : 2 \\ & \mathbf{ a } & \mathbf{=} & \mathbf{41} \\ \hline \end{array}\)

The first term in the sequence **41**

heureka
Aug 3, 2018