insert two arithmetic means between 5 and 19
Answer (1)
SOLUTION:Step 1: List the given values.Since we must insert 2 arithmetic means between 5 and 19, the number of terms (2 arithmetic means + 2 given terms) is 4.[tex]\begin{aligned} & a_1 = 5 \\ & a_4 = 19 \\ & n = 4 \end{aligned}[/tex]Step 2: Determine the common difference.[tex]\begin{aligned} a_n & = a_1 + (n - 1)d \\ a_4 & = 5 + (4 - 1)d \\ 19 & = 5 + 3d \\ 5 + 3d & = 19 \\ 3d & = 19 - 5 \\ 3d & = 14 \\ \frac{3d}{3} & = \frac{14}{3} \\ d & = \frac{14}{3} \end{aligned}[/tex]Step 3: Solve for the 2 arithmetic means.• For a₂[tex]\begin{aligned} a_2 & = a_1 + (2 - 1)d \\ a_2 & = a_1 + d \\ a_2 & = 5 + \frac{14}{3} \\ & = \boxed{\frac{29}{3}} \end{aligned}[/tex]• For a₃[tex]\begin{aligned} a_3 & = a_1 + (3 - 1)d \\ a_3 & = a_1 + 2d \\ a_3 & = 5 + 2\left(\frac{14}{3}\right) \\ a_3 & = 5 + \frac{28}{3} \\ & = \boxed{\frac{43}{3}} \end{aligned}[/tex]Hence, the 2 arithmetic means between 5 and 19 are 29/3 and 43/3, respectively.
