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Forgot how to do this......

Find the fourth arithmetic means between -21 and -36

Apr 5, 2018

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Find the fourth arithmetic means between -21 and -36

$$\text{We use a_n = a_1 + (n-1)d to find the common difference d.}$$

Solution

$$\text{The first term is a_1 = - 21 and the fourth term is a_4 = - 36 .}\\ \text{We must find the common difference so that the terms }$$

$$\begin{array}{cccc} -21, & -21+d, & -21+2d, & -36 \\ \uparrow & \uparrow & \uparrow & \uparrow \\ a_1 & a_2 & a_3 & a_4 \end{array}$$

$$\text{form an arithmetic sequence.}$$

$$\text{To find the common difference d, we substitue -21 for a_1; 4 for n, and -36 for a_n \\in the formula for the 4th term:}$$

$$\begin{array}{rcll} a_4 &=& a_1 + (n-1)d \qquad & \text{This gives the 4th term of any arithmetic sequence.} \\ -36 &=& -21 + (4-1)d \qquad & \text{Substitute.} \\ -36 &=& -21 + 3d \qquad & \text{Subtract within the parentheses.}\\ -15 &=& 3d \qquad & \text{Subtract -21 from both sides.} \\ -5 &=& d \qquad & \text{To isolate d, divide both sides by 3.} \\ \end{array}$$

$$\text{To find the two arithmetic means between -21 and -36, \\we add the common difference -5, as shown:}$$

$$\begin{array}{rcll} -21+d &=& -21 +(-5) \\ &=& -26 \qquad & \text{This is a_2.} \\ \end{array}$$

$$\begin{array}{rcll} -21+2d &=& -21 +2(-5) \\ &=& -21 -10 \\ &=& -31 \qquad & \text{This is a_3.} \\ \end{array}$$

$$\text{Two arithmetic means between -21 and -36 are -26 and -31.}$$

Apr 5, 2018