insert three arithmetic means between 2 and 22
Answer (1)
SOLUTION:Step 1: List the given values.Since we must insert 3 arithmetic means between 2 and 22, the number of terms (3 arithmetic means + 2 given terms) is 5.[tex]\begin{aligned} & a_1 = 2 \\ & a_5 = 22 \\ & n = 5 \end{aligned}[/tex]Step 2: Determine the common difference.[tex]\begin{aligned} a_n & = a_1 + (n - 1)d \\ a_5 & = 2 + (5 - 1)d \\ 22 & = 2 + 4d \\ 2 + 4d & = 22 \\ 4d & = 22 - 2 \\ 4d & = 20 \\ \frac{4d}{4} & = \frac{20}{4} \\ d & = 5 \end{aligned}[/tex]Step 3: Solve for the 3 arithmetic means.• For a₂[tex]\begin{aligned} a_2 & = a_1 + (2 - 1)d \\ a_2 & = a_1 + d \\ a_2 & = 2 + 5 \\ & = \boxed{7} \end{aligned}[/tex]• For a₃[tex]\begin{aligned} a_3 & = a_1 + (3 - 1)d \\ a_3 & = a_1 + 2d \\ a_3 & = 2 + 2(5) \\ a_3 & = 2 + 10 \\ & = \boxed{12} \end{aligned}[/tex]• For a₄[tex]\begin{aligned} a_4 & = a_1 + (4 - 1)d \\ a_4 & = a_1 + 3d \\ a_4 & = 2 + 3(5) \\ a_4 & = 2 + 15 \\ & = \boxed{17} \end{aligned}[/tex]Hence, the 3 arithmetic means between 2 and 22 are 7, 12, and 17, respectively.
