Arithmetic means are the missing terms placed between two given numbers so that the entire list forms an arithmetic sequence. To find them, first count the total number of terms, including the two endpoints, and then use the arithmetic sequence formula to find the common difference. These problems practice inserting one or more arithmetic means between two terms, including positive numbers, negative numbers, fractions, and decimals.
Practice Problems
\(\textbf{1)}\) Find the three arithmetic means of \(3\) and \(48\)
The answer is \(\frac{57}{4},\frac{51}{2},\frac{147}{4}\)
\(\,\,\,\,\,\,\,\,\,3,\, \text{___},\, \text{___},\, \text{___},\, 48\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,48=3+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,48=3+4d\)
\(\,\,\,\,\,\,\,\,\,45=4d\)
\(\,\,\,\,\,\,\,\,\,d=\frac{45}{4}\)
\(\,\,\,\,\,\,\,\,\,3,\, \underline{3+\frac{45}{4}},\, \underline{3+\frac{90}{4}},\, \underline{3+\frac{135}{4}},\, 48\)
\(\,\,\,\,\,\,\,\,\,3,\, \underline{\frac{57}{4}},\, \underline{\frac{51}{2}},\, \underline{\frac{147}{4}},\, 48\)
\(\textbf{2)}\) Find the four arithmetic means of \(56\) and \(356\)
\(\,\,\,\)The answer is \(116,176,236,296\)
\(\,\,\,\,\,\,\,\,\,56,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 356\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,356=56+d(6-1)\)
\(\,\,\,\,\,\,\,\,\,356=56+5d\)
\(\,\,\,\,\,\,\,\,\,300=5d\)
\(\,\,\,\,\,\,\,\,\,60=d\)
\(\,\,\,\,\,\,\,\,\,56,\, \underline{56+60},\, \underline{56+120},\, \underline{56+180},\, \underline{56+240},\, 356\)
\(\,\,\,\,\,\,\,\,\,56,\, \underline{116},\, \underline{176},\, \underline{236},\, \underline{296},\, 356\)
\(\textbf{3)}\) Find the three arithmetic means of \(2\) and \(10\)
\(\,\,\,\)The answer is \(4,6,8\)
\(\,\,\,\,\,\,\,\,\,2,\, \text{___},\, \text{___},\, \text{___},\, 10\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,10=2+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,10=2+4d\)
\(\,\,\,\,\,\,\,\,\,8=4d\)
\(\,\,\,\,\,\,\,\,\,2=d\)
\(\,\,\,\,\,\,\,\,\,2,\, \underline{2+2},\, \underline{2+4},\, \underline{2+6},\, 10\)
\(\,\,\,\,\,\,\,\,\,2,\, \underline{4},\, \underline{6},\, \underline{8},\, 10\)
\(\textbf{4)}\) Find the two arithmetic means of \(-5\) and \(40\)
\(\,\,\,\)The answer is \(10,25\)
\(\,\,\,\,\,\,\,\,\,-5,\, \text{___},\, \text{___},\, 40\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,40=-5+d(4-1)\)
\(\,\,\,\,\,\,\,\,\,40=-5+3d\)
\(\,\,\,\,\,\,\,\,\,45=3d\)
\(\,\,\,\,\,\,\,\,\,15=d\)
\(\,\,\,\,\,\,\,\,\,-5,\, \underline{-5+15},\, \underline{-5+30},\, 40\)
\(\,\,\,\,\,\,\,\,\,-5,\, \underline{10},\, \underline{25},\, 40\)
\(\textbf{5)}\) Find the arithmetic mean of \(8\) and \(20\)
\(\,\,\,\)The answer is \(14\)
\(\,\,\,\,\,\,\,\,\,8,\, \text{___},\, 20\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,20=8+d(3-1)\)
\(\,\,\,\,\,\,\,\,\,20=8+2d\)
\(\,\,\,\,\,\,\,\,\,12=2d\)
\(\,\,\,\,\,\,\,\,\,6=d\)
\(\,\,\,\,\,\,\,\,\,8,\, \underline{8+6},\, 20\)
\(\,\,\,\,\,\,\,\,\,8,\, \underline{14},\, 20\)
\(\textbf{6)}\) Find the two arithmetic means of \(4\) and \(25\)
\(\,\,\,\)The answer is \(11,18\)
\(\,\,\,\,\,\,\,\,\,4,\, \text{___},\, \text{___},\, 25\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,25=4+d(4-1)\)
\(\,\,\,\,\,\,\,\,\,25=4+3d\)
\(\,\,\,\,\,\,\,\,\,21=3d\)
\(\,\,\,\,\,\,\,\,\,7=d\)
\(\,\,\,\,\,\,\,\,\,4,\, \underline{4+7},\, \underline{4+14},\, 25\)
\(\,\,\,\,\,\,\,\,\,4,\, \underline{11},\, \underline{18},\, 25\)
\(\textbf{7)}\) Find the three arithmetic means of \(-12\) and \(8\)
\(\,\,\,\)The answer is \(-7,-2,3\)
\(\,\,\,\,\,\,\,\,\,-12,\, \text{___},\, \text{___},\, \text{___},\, 8\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,8=-12+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,8=-12+4d\)
\(\,\,\,\,\,\,\,\,\,20=4d\)
\(\,\,\,\,\,\,\,\,\,5=d\)
\(\,\,\,\,\,\,\,\,\,-12,\, \underline{-12+5},\, \underline{-12+10},\, \underline{-12+15},\, 8\)
\(\,\,\,\,\,\,\,\,\,-12,\, \underline{-7},\, \underline{-2},\, \underline{3},\, 8\)
\(\textbf{8)}\) Find the four arithmetic means of \(10\) and \(-15\)
\(\,\,\,\)The answer is \(5,0,-5,-10\)
\(\,\,\,\,\,\,\,\,\,10,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, -15\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,-15=10+d(6-1)\)
\(\,\,\,\,\,\,\,\,\,-15=10+5d\)
\(\,\,\,\,\,\,\,\,\,-25=5d\)
\(\,\,\,\,\,\,\,\,\,-5=d\)
\(\,\,\,\,\,\,\,\,\,10,\, \underline{10-5},\, \underline{10-10},\, \underline{10-15},\, \underline{10-20},\, -15\)
\(\,\,\,\,\,\,\,\,\,10,\, \underline{5},\, \underline{0},\, \underline{-5},\, \underline{-10},\, -15\)
\(\textbf{9)}\) Find the five arithmetic means of \(3\) and \(21\)
\(\,\,\,\)The answer is \(6,9,12,15,18\)
\(\,\,\,\,\,\,\,\,\,3,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 21\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,21=3+d(7-1)\)
\(\,\,\,\,\,\,\,\,\,21=3+6d\)
\(\,\,\,\,\,\,\,\,\,18=6d\)
\(\,\,\,\,\,\,\,\,\,3=d\)
\(\,\,\,\,\,\,\,\,\,3,\, \underline{6},\, \underline{9},\, \underline{12},\, \underline{15},\, \underline{18},\, 21\)
\(\textbf{10)}\) Find the two arithmetic means of \(\frac{1}{2}\) and \(\frac{7}{2}\)
\(\,\,\,\)The answer is \(\frac{3}{2},\frac{5}{2}\)
\(\,\,\,\,\,\,\,\,\,\frac{1}{2},\, \text{___},\, \text{___},\, \frac{7}{2}\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,\frac{7}{2}=\frac{1}{2}+d(4-1)\)
\(\,\,\,\,\,\,\,\,\,\frac{7}{2}=\frac{1}{2}+3d\)
\(\,\,\,\,\,\,\,\,\,3=3d\)
\(\,\,\,\,\,\,\,\,\,1=d\)
\(\,\,\,\,\,\,\,\,\,\frac{1}{2},\, \underline{\frac{1}{2}+1},\, \underline{\frac{1}{2}+2},\, \frac{7}{2}\)
\(\,\,\,\,\,\,\,\,\,\frac{1}{2},\, \underline{\frac{3}{2}},\, \underline{\frac{5}{2}},\, \frac{7}{2}\)
\(\textbf{11)}\) Find the three arithmetic means of \(2.5\) and \(12.5\)
\(\,\,\,\)The answer is \(5,7.5,10\)
\(\,\,\,\,\,\,\,\,\,2.5,\, \text{___},\, \text{___},\, \text{___},\, 12.5\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,12.5=2.5+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,12.5=2.5+4d\)
\(\,\,\,\,\,\,\,\,\,10=4d\)
\(\,\,\,\,\,\,\,\,\,2.5=d\)
\(\,\,\,\,\,\,\,\,\,2.5,\, \underline{5},\, \underline{7.5},\, \underline{10},\, 12.5\)
\(\textbf{12)}\) Find the four arithmetic means of \(-2\) and \(18\)
\(\,\,\,\)The answer is \(2,6,10,14\)
\(\,\,\,\,\,\,\,\,\,-2,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 18\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,18=-2+d(6-1)\)
\(\,\,\,\,\,\,\,\,\,18=-2+5d\)
\(\,\,\,\,\,\,\,\,\,20=5d\)
\(\,\,\,\,\,\,\,\,\,4=d\)
\(\,\,\,\,\,\,\,\,\,-2,\, \underline{2},\, \underline{6},\, \underline{10},\, \underline{14},\, 18\)
\(\textbf{13)}\) Find the arithmetic mean of \(-7\) and \(9\)
\(\,\,\,\)The answer is \(1\)
\(\,\,\,\,\,\,\,\,\,-7,\, \text{___},\, 9\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,9=-7+d(3-1)\)
\(\,\,\,\,\,\,\,\,\,9=-7+2d\)
\(\,\,\,\,\,\,\,\,\,16=2d\)
\(\,\,\,\,\,\,\,\,\,8=d\)
\(\,\,\,\,\,\,\,\,\,-7,\, \underline{-7+8},\, 9\)
\(\,\,\,\,\,\,\,\,\,-7,\, \underline{1},\, 9\)
\(\textbf{14)}\) Find the two arithmetic means of \(100\) and \(40\)
\(\,\,\,\)The answer is \(80,60\)
\(\,\,\,\,\,\,\,\,\,100,\, \text{___},\, \text{___},\, 40\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,40=100+d(4-1)\)
\(\,\,\,\,\,\,\,\,\,40=100+3d\)
\(\,\,\,\,\,\,\,\,\,-60=3d\)
\(\,\,\,\,\,\,\,\,\,-20=d\)
\(\,\,\,\,\,\,\,\,\,100,\, \underline{100-20},\, \underline{100-40},\, 40\)
\(\,\,\,\,\,\,\,\,\,100,\, \underline{80},\, \underline{60},\, 40\)
\(\textbf{15)}\) Find the three arithmetic means of \(\frac{1}{3}\) and \(\frac{13}{3}\)
\(\,\,\,\)The answer is \(\frac{4}{3},\frac{7}{3},\frac{10}{3}\)
\(\,\,\,\,\,\,\,\,\,\frac{1}{3},\, \text{___},\, \text{___},\, \text{___},\, \frac{13}{3}\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,\frac{13}{3}=\frac{1}{3}+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,\frac{13}{3}=\frac{1}{3}+4d\)
\(\,\,\,\,\,\,\,\,\,4=4d\)
\(\,\,\,\,\,\,\,\,\,1=d\)
\(\,\,\,\,\,\,\,\,\,\frac{1}{3},\, \underline{\frac{4}{3}},\, \underline{\frac{7}{3}},\, \underline{\frac{10}{3}},\, \frac{13}{3}\)
\(\textbf{16)}\) Find the five arithmetic means of \(-9\) and \(33\)
\(\,\,\,\)The answer is \(-2,5,12,19,26\)
\(\,\,\,\,\,\,\,\,\,-9,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 33\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,33=-9+d(7-1)\)
\(\,\,\,\,\,\,\,\,\,33=-9+6d\)
\(\,\,\,\,\,\,\,\,\,42=6d\)
\(\,\,\,\,\,\,\,\,\,7=d\)
\(\,\,\,\,\,\,\,\,\,-9,\, \underline{-2},\, \underline{5},\, \underline{12},\, \underline{19},\, \underline{26},\, 33\)
\(\textbf{17)}\) Find the four arithmetic means of \(5\) and \(35\)
\(\,\,\,\)The answer is \(11,17,23,29\)
\(\,\,\,\,\,\,\,\,\,5,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 35\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,35=5+d(6-1)\)
\(\,\,\,\,\,\,\,\,\,35=5+5d\)
\(\,\,\,\,\,\,\,\,\,30=5d\)
\(\,\,\,\,\,\,\,\,\,6=d\)
\(\,\,\,\,\,\,\,\,\,5,\, \underline{11},\, \underline{17},\, \underline{23},\, \underline{29},\, 35\)
\(\textbf{18)}\) Find the two arithmetic means of \(-1.5\) and \(7.5\)
\(\,\,\,\)The answer is \(1.5,4.5\)
\(\,\,\,\,\,\,\,\,\,-1.5,\, \text{___},\, \text{___},\, 7.5\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,7.5=-1.5+d(4-1)\)
\(\,\,\,\,\,\,\,\,\,7.5=-1.5+3d\)
\(\,\,\,\,\,\,\,\,\,9=3d\)
\(\,\,\,\,\,\,\,\,\,3=d\)
\(\,\,\,\,\,\,\,\,\,-1.5,\, \underline{-1.5+3},\, \underline{-1.5+6},\, 7.5\)
\(\,\,\,\,\,\,\,\,\,-1.5,\, \underline{1.5},\, \underline{4.5},\, 7.5\)
\(\textbf{19)}\) Find the three arithmetic means of \(0\) and \(100\)
\(\,\,\,\)The answer is \(25,50,75\)
\(\,\,\,\,\,\,\,\,\,0,\, \text{___},\, \text{___},\, \text{___},\, 100\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,100=0+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,100=4d\)
\(\,\,\,\,\,\,\,\,\,25=d\)
\(\,\,\,\,\,\,\,\,\,0,\, \underline{25},\, \underline{50},\, \underline{75},\, 100\)
\(\textbf{20)}\) How many arithmetic means are inserted between \(4\) and \(34\) if the common difference is \(5\)?
\(\,\,\,\)The answer is \(5\) arithmetic means
\(\,\,\,\,\,\,\,\,\,4,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 34\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,34=4+5(n-1)\)
\(\,\,\,\,\,\,\,\,\,30=5(n-1)\)
\(\,\,\,\,\,\,\,\,\,6=n-1\)
\(\,\,\,\,\,\,\,\,\,7=n\)
\(\,\,\,\,\,\,\,\,\,\text{There are }7\text{ total terms.}\)
\(\,\,\,\,\,\,\,\,\,7-2=5\text{ arithmetic means}\)
\(\,\,\,\,\,\,\,\,\,4,\,9,\,14,\,19,\,24,\,29,\,34\)
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