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 \(\frac{57}{4},\frac{51}{2},\frac{147}{4}\)
\(\,\,\,\,\,\,\,\,\,3,\, \text{___},\, \text{___},\, \text{___},\, 48\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,48=3+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,48=3+4d\)
\(\,\,\,\,\,\,\,\,\,45=4d\)
\(\,\,\,\,\,\,\,\,\,d=\frac{45}{4}\)
\(\,\,\,\,\,\,\,\,\,3,\, \underline{3+\frac{45}{4}},\, \underline{3+\frac{90}{4}},\, \underline{3+\frac{135}{4}},\, 48\)
\(\,\,\,\,\,\,\,\,\,3,\, \underline{\frac{57}{4}},\, \underline{\frac{51}{2}},\, \underline{\frac{147}{4}},\, 48\)
\(\,\,\,\,\,\,\,\,\,3,\, \text{___},\, \text{___},\, \text{___},\, 48\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,48=3+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,48=3+4d\)
\(\,\,\,\,\,\,\,\,\,45=4d\)
\(\,\,\,\,\,\,\,\,\,d=\frac{45}{4}\)
\(\,\,\,\,\,\,\,\,\,3,\, \underline{3+\frac{45}{4}},\, \underline{3+\frac{90}{4}},\, \underline{3+\frac{135}{4}},\, 48\)
\(\,\,\,\,\,\,\,\,\,3,\, \underline{\frac{57}{4}},\, \underline{\frac{51}{2}},\, \underline{\frac{147}{4}},\, 48\)
\(\textbf{2)}\) Find the four arithmetic means of \(56\) and \(356\)\(\,\,\,\)The answer is \(116,176,236,296\)
\(\,\,\,\,\,\,\,\,\,56,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 356\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,356=56+d(6-1)\)
\(\,\,\,\,\,\,\,\,\,356=56+5d\)
\(\,\,\,\,\,\,\,\,\,300=5d\)
\(\,\,\,\,\,\,\,\,\,60=d\)
\(\,\,\,\,\,\,\,\,\,56,\, \underline{56+60},\, \underline{56+120},\, \underline{56+180},\, \underline{56+240},\, 356\)
\(\,\,\,\,\,\,\,\,\,56,\, \underline{116},\, \underline{176},\, \underline{236},\, \underline{296},\, 356\)
\(\,\,\,\,\,\,\,\,\,56,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 356\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,356=56+d(6-1)\)
\(\,\,\,\,\,\,\,\,\,356=56+5d\)
\(\,\,\,\,\,\,\,\,\,300=5d\)
\(\,\,\,\,\,\,\,\,\,60=d\)
\(\,\,\,\,\,\,\,\,\,56,\, \underline{56+60},\, \underline{56+120},\, \underline{56+180},\, \underline{56+240},\, 356\)
\(\,\,\,\,\,\,\,\,\,56,\, \underline{116},\, \underline{176},\, \underline{236},\, \underline{296},\, 356\)
\(\textbf{3)}\) Find the three arithmetic means of \(2\) and \(10\)\(\,\,\,\)The answer is \(4,6,8\)
\(\,\,\,\,\,\,\,\,\,2,\, \text{___},\, \text{___},\, \text{___},\, 10\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,10=2+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,10=2+4d\)
\(\,\,\,\,\,\,\,\,\,8=4d\)
\(\,\,\,\,\,\,\,\,\,2=d\)
\(\,\,\,\,\,\,\,\,\,2,\, \underline{2+2},\, \underline{2+4},\, \underline{2+6},\, 10\)
\(\,\,\,\,\,\,\,\,\,2,\, \underline{4},\, \underline{6},\, \underline{8},\, 10\)
\(\,\,\,\,\,\,\,\,\,2,\, \text{___},\, \text{___},\, \text{___},\, 10\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,10=2+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,10=2+4d\)
\(\,\,\,\,\,\,\,\,\,8=4d\)
\(\,\,\,\,\,\,\,\,\,2=d\)
\(\,\,\,\,\,\,\,\,\,2,\, \underline{2+2},\, \underline{2+4},\, \underline{2+6},\, 10\)
\(\,\,\,\,\,\,\,\,\,2,\, \underline{4},\, \underline{6},\, \underline{8},\, 10\)
\(\textbf{4)}\) Find the two arithmetic means of \(-5\) and \(40\)\(\,\,\,\)The answer is \(10,25\)
\(\,\,\,\,\,\,\,\,\,-5,\, \text{___},\, \text{___},\, 40\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,40=-5+d(4-1)\)
\(\,\,\,\,\,\,\,\,\,40=-5+3d\)
\(\,\,\,\,\,\,\,\,\,45=3d\)
\(\,\,\,\,\,\,\,\,\,15=d\)
\(\,\,\,\,\,\,\,\,\,-5,\, \underline{-5+15},\, \underline{-5+30},\, 40\)
\(\,\,\,\,\,\,\,\,\,-5,\, \underline{10},\, \underline{25},\, 40\)
\(\,\,\,\,\,\,\,\,\,-5,\, \text{___},\, \text{___},\, 40\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,40=-5+d(4-1)\)
\(\,\,\,\,\,\,\,\,\,40=-5+3d\)
\(\,\,\,\,\,\,\,\,\,45=3d\)
\(\,\,\,\,\,\,\,\,\,15=d\)
\(\,\,\,\,\,\,\,\,\,-5,\, \underline{-5+15},\, \underline{-5+30},\, 40\)
\(\,\,\,\,\,\,\,\,\,-5,\, \underline{10},\, \underline{25},\, 40\)
\(\textbf{5)}\) Find the arithmetic mean of \(8\) and \(20\)\(\,\,\,\)The answer is \(14\)
\(\,\,\,\,\,\,\,\,\,8,\, \text{___},\, 20\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,20=8+d(3-1)\)
\(\,\,\,\,\,\,\,\,\,20=8+2d\)
\(\,\,\,\,\,\,\,\,\,12=2d\)
\(\,\,\,\,\,\,\,\,\,6=d\)
\(\,\,\,\,\,\,\,\,\,8,\, \underline{8+6},\, 20\)
\(\,\,\,\,\,\,\,\,\,8,\, \underline{14},\, 20\)
\(\,\,\,\,\,\,\,\,\,8,\, \text{___},\, 20\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,20=8+d(3-1)\)
\(\,\,\,\,\,\,\,\,\,20=8+2d\)
\(\,\,\,\,\,\,\,\,\,12=2d\)
\(\,\,\,\,\,\,\,\,\,6=d\)
\(\,\,\,\,\,\,\,\,\,8,\, \underline{8+6},\, 20\)
\(\,\,\,\,\,\,\,\,\,8,\, \underline{14},\, 20\)
\(\textbf{6)}\) Find the two arithmetic means of \(4\) and \(25\)\(\,\,\,\)The answer is \(11,18\)
\(\,\,\,\,\,\,\,\,\,4,\, \text{___},\, \text{___},\, 25\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,25=4+d(4-1)\)
\(\,\,\,\,\,\,\,\,\,25=4+3d\)
\(\,\,\,\,\,\,\,\,\,21=3d\)
\(\,\,\,\,\,\,\,\,\,7=d\)
\(\,\,\,\,\,\,\,\,\,4,\, \underline{4+7},\, \underline{4+14},\, 25\)
\(\,\,\,\,\,\,\,\,\,4,\, \underline{11},\, \underline{18},\, 25\)
\(\,\,\,\,\,\,\,\,\,4,\, \text{___},\, \text{___},\, 25\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,25=4+d(4-1)\)
\(\,\,\,\,\,\,\,\,\,25=4+3d\)
\(\,\,\,\,\,\,\,\,\,21=3d\)
\(\,\,\,\,\,\,\,\,\,7=d\)
\(\,\,\,\,\,\,\,\,\,4,\, \underline{4+7},\, \underline{4+14},\, 25\)
\(\,\,\,\,\,\,\,\,\,4,\, \underline{11},\, \underline{18},\, 25\)
\(\textbf{7)}\) Find the three arithmetic means of \(-12\) and \(8\)\(\,\,\,\)The answer is \(-7,-2,3\)
\(\,\,\,\,\,\,\,\,\,-12,\, \text{___},\, \text{___},\, \text{___},\, 8\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,8=-12+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,8=-12+4d\)
\(\,\,\,\,\,\,\,\,\,20=4d\)
\(\,\,\,\,\,\,\,\,\,5=d\)
\(\,\,\,\,\,\,\,\,\,-12,\, \underline{-12+5},\, \underline{-12+10},\, \underline{-12+15},\, 8\)
\(\,\,\,\,\,\,\,\,\,-12,\, \underline{-7},\, \underline{-2},\, \underline{3},\, 8\)
\(\,\,\,\,\,\,\,\,\,-12,\, \text{___},\, \text{___},\, \text{___},\, 8\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,8=-12+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,8=-12+4d\)
\(\,\,\,\,\,\,\,\,\,20=4d\)
\(\,\,\,\,\,\,\,\,\,5=d\)
\(\,\,\,\,\,\,\,\,\,-12,\, \underline{-12+5},\, \underline{-12+10},\, \underline{-12+15},\, 8\)
\(\,\,\,\,\,\,\,\,\,-12,\, \underline{-7},\, \underline{-2},\, \underline{3},\, 8\)
\(\textbf{8)}\) Find the four arithmetic means of \(10\) and \(-15\)\(\,\,\,\)The answer is \(5,0,-5,-10\)
\(\,\,\,\,\,\,\,\,\,10,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, -15\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,-15=10+d(6-1)\)
\(\,\,\,\,\,\,\,\,\,-15=10+5d\)
\(\,\,\,\,\,\,\,\,\,-25=5d\)
\(\,\,\,\,\,\,\,\,\,-5=d\)
\(\,\,\,\,\,\,\,\,\,10,\, \underline{10-5},\, \underline{10-10},\, \underline{10-15},\, \underline{10-20},\, -15\)
\(\,\,\,\,\,\,\,\,\,10,\, \underline{5},\, \underline{0},\, \underline{-5},\, \underline{-10},\, -15\)
\(\,\,\,\,\,\,\,\,\,10,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, -15\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,-15=10+d(6-1)\)
\(\,\,\,\,\,\,\,\,\,-15=10+5d\)
\(\,\,\,\,\,\,\,\,\,-25=5d\)
\(\,\,\,\,\,\,\,\,\,-5=d\)
\(\,\,\,\,\,\,\,\,\,10,\, \underline{10-5},\, \underline{10-10},\, \underline{10-15},\, \underline{10-20},\, -15\)
\(\,\,\,\,\,\,\,\,\,10,\, \underline{5},\, \underline{0},\, \underline{-5},\, \underline{-10},\, -15\)
\(\textbf{9)}\) Find the five arithmetic means of \(3\) and \(21\)\(\,\,\,\)The answer is \(6,9,12,15,18\)
\(\,\,\,\,\,\,\,\,\,3,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 21\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,21=3+d(7-1)\)
\(\,\,\,\,\,\,\,\,\,21=3+6d\)
\(\,\,\,\,\,\,\,\,\,18=6d\)
\(\,\,\,\,\,\,\,\,\,3=d\)
\(\,\,\,\,\,\,\,\,\,3,\, \underline{6},\, \underline{9},\, \underline{12},\, \underline{15},\, \underline{18},\, 21\)
\(\,\,\,\,\,\,\,\,\,3,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 21\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,21=3+d(7-1)\)
\(\,\,\,\,\,\,\,\,\,21=3+6d\)
\(\,\,\,\,\,\,\,\,\,18=6d\)
\(\,\,\,\,\,\,\,\,\,3=d\)
\(\,\,\,\,\,\,\,\,\,3,\, \underline{6},\, \underline{9},\, \underline{12},\, \underline{15},\, \underline{18},\, 21\)
\(\textbf{10)}\) Find the two arithmetic means of \(\frac{1}{2}\) and \(\frac{7}{2}\)\(\,\,\,\)The answer is \(\frac{3}{2},\frac{5}{2}\)
\(\,\,\,\,\,\,\,\,\,\frac{1}{2},\, \text{___},\, \text{___},\, \frac{7}{2}\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,\frac{7}{2}=\frac{1}{2}+d(4-1)\)
\(\,\,\,\,\,\,\,\,\,\frac{7}{2}=\frac{1}{2}+3d\)
\(\,\,\,\,\,\,\,\,\,3=3d\)
\(\,\,\,\,\,\,\,\,\,1=d\)
\(\,\,\,\,\,\,\,\,\,\frac{1}{2},\, \underline{\frac{1}{2}+1},\, \underline{\frac{1}{2}+2},\, \frac{7}{2}\)
\(\,\,\,\,\,\,\,\,\,\frac{1}{2},\, \underline{\frac{3}{2}},\, \underline{\frac{5}{2}},\, \frac{7}{2}\)
\(\,\,\,\,\,\,\,\,\,\frac{1}{2},\, \text{___},\, \text{___},\, \frac{7}{2}\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,\frac{7}{2}=\frac{1}{2}+d(4-1)\)
\(\,\,\,\,\,\,\,\,\,\frac{7}{2}=\frac{1}{2}+3d\)
\(\,\,\,\,\,\,\,\,\,3=3d\)
\(\,\,\,\,\,\,\,\,\,1=d\)
\(\,\,\,\,\,\,\,\,\,\frac{1}{2},\, \underline{\frac{1}{2}+1},\, \underline{\frac{1}{2}+2},\, \frac{7}{2}\)
\(\,\,\,\,\,\,\,\,\,\frac{1}{2},\, \underline{\frac{3}{2}},\, \underline{\frac{5}{2}},\, \frac{7}{2}\)
\(\textbf{11)}\) Find the three arithmetic means of \(2.5\) and \(12.5\)\(\,\,\,\)The answer is \(5,7.5,10\)
\(\,\,\,\,\,\,\,\,\,2.5,\, \text{___},\, \text{___},\, \text{___},\, 12.5\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,12.5=2.5+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,12.5=2.5+4d\)
\(\,\,\,\,\,\,\,\,\,10=4d\)
\(\,\,\,\,\,\,\,\,\,2.5=d\)
\(\,\,\,\,\,\,\,\,\,2.5,\, \underline{5},\, \underline{7.5},\, \underline{10},\, 12.5\)
\(\,\,\,\,\,\,\,\,\,2.5,\, \text{___},\, \text{___},\, \text{___},\, 12.5\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,12.5=2.5+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,12.5=2.5+4d\)
\(\,\,\,\,\,\,\,\,\,10=4d\)
\(\,\,\,\,\,\,\,\,\,2.5=d\)
\(\,\,\,\,\,\,\,\,\,2.5,\, \underline{5},\, \underline{7.5},\, \underline{10},\, 12.5\)
\(\textbf{12)}\) Find the four arithmetic means of \(-2\) and \(18\)\(\,\,\,\)The answer is \(2,6,10,14\)
\(\,\,\,\,\,\,\,\,\,-2,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 18\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,18=-2+d(6-1)\)
\(\,\,\,\,\,\,\,\,\,18=-2+5d\)
\(\,\,\,\,\,\,\,\,\,20=5d\)
\(\,\,\,\,\,\,\,\,\,4=d\)
\(\,\,\,\,\,\,\,\,\,-2,\, \underline{2},\, \underline{6},\, \underline{10},\, \underline{14},\, 18\)
\(\,\,\,\,\,\,\,\,\,-2,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 18\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,18=-2+d(6-1)\)
\(\,\,\,\,\,\,\,\,\,18=-2+5d\)
\(\,\,\,\,\,\,\,\,\,20=5d\)
\(\,\,\,\,\,\,\,\,\,4=d\)
\(\,\,\,\,\,\,\,\,\,-2,\, \underline{2},\, \underline{6},\, \underline{10},\, \underline{14},\, 18\)
\(\textbf{13)}\) Find the arithmetic mean of \(-7\) and \(9\)\(\,\,\,\)The answer is \(1\)
\(\,\,\,\,\,\,\,\,\,-7,\, \text{___},\, 9\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,9=-7+d(3-1)\)
\(\,\,\,\,\,\,\,\,\,9=-7+2d\)
\(\,\,\,\,\,\,\,\,\,16=2d\)
\(\,\,\,\,\,\,\,\,\,8=d\)
\(\,\,\,\,\,\,\,\,\,-7,\, \underline{-7+8},\, 9\)
\(\,\,\,\,\,\,\,\,\,-7,\, \underline{1},\, 9\)
\(\,\,\,\,\,\,\,\,\,-7,\, \text{___},\, 9\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,9=-7+d(3-1)\)
\(\,\,\,\,\,\,\,\,\,9=-7+2d\)
\(\,\,\,\,\,\,\,\,\,16=2d\)
\(\,\,\,\,\,\,\,\,\,8=d\)
\(\,\,\,\,\,\,\,\,\,-7,\, \underline{-7+8},\, 9\)
\(\,\,\,\,\,\,\,\,\,-7,\, \underline{1},\, 9\)
\(\textbf{14)}\) Find the two arithmetic means of \(100\) and \(40\)\(\,\,\,\)The answer is \(80,60\)
\(\,\,\,\,\,\,\,\,\,100,\, \text{___},\, \text{___},\, 40\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,40=100+d(4-1)\)
\(\,\,\,\,\,\,\,\,\,40=100+3d\)
\(\,\,\,\,\,\,\,\,\,-60=3d\)
\(\,\,\,\,\,\,\,\,\,-20=d\)
\(\,\,\,\,\,\,\,\,\,100,\, \underline{100-20},\, \underline{100-40},\, 40\)
\(\,\,\,\,\,\,\,\,\,100,\, \underline{80},\, \underline{60},\, 40\)
\(\,\,\,\,\,\,\,\,\,100,\, \text{___},\, \text{___},\, 40\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,40=100+d(4-1)\)
\(\,\,\,\,\,\,\,\,\,40=100+3d\)
\(\,\,\,\,\,\,\,\,\,-60=3d\)
\(\,\,\,\,\,\,\,\,\,-20=d\)
\(\,\,\,\,\,\,\,\,\,100,\, \underline{100-20},\, \underline{100-40},\, 40\)
\(\,\,\,\,\,\,\,\,\,100,\, \underline{80},\, \underline{60},\, 40\)
\(\textbf{15)}\) Find the three arithmetic means of \(\frac{1}{3}\) and \(\frac{13}{3}\)\(\,\,\,\)The answer is \(\frac{4}{3},\frac{7}{3},\frac{10}{3}\)
\(\,\,\,\,\,\,\,\,\,\frac{1}{3},\, \text{___},\, \text{___},\, \text{___},\, \frac{13}{3}\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,\frac{13}{3}=\frac{1}{3}+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,\frac{13}{3}=\frac{1}{3}+4d\)
\(\,\,\,\,\,\,\,\,\,4=4d\)
\(\,\,\,\,\,\,\,\,\,1=d\)
\(\,\,\,\,\,\,\,\,\,\frac{1}{3},\, \underline{\frac{4}{3}},\, \underline{\frac{7}{3}},\, \underline{\frac{10}{3}},\, \frac{13}{3}\)
\(\,\,\,\,\,\,\,\,\,\frac{1}{3},\, \text{___},\, \text{___},\, \text{___},\, \frac{13}{3}\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,\frac{13}{3}=\frac{1}{3}+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,\frac{13}{3}=\frac{1}{3}+4d\)
\(\,\,\,\,\,\,\,\,\,4=4d\)
\(\,\,\,\,\,\,\,\,\,1=d\)
\(\,\,\,\,\,\,\,\,\,\frac{1}{3},\, \underline{\frac{4}{3}},\, \underline{\frac{7}{3}},\, \underline{\frac{10}{3}},\, \frac{13}{3}\)
\(\textbf{16)}\) Find the five arithmetic means of \(-9\) and \(33\)\(\,\,\,\)The answer is \(-2,5,12,19,26\)
\(\,\,\,\,\,\,\,\,\,-9,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 33\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,33=-9+d(7-1)\)
\(\,\,\,\,\,\,\,\,\,33=-9+6d\)
\(\,\,\,\,\,\,\,\,\,42=6d\)
\(\,\,\,\,\,\,\,\,\,7=d\)
\(\,\,\,\,\,\,\,\,\,-9,\, \underline{-2},\, \underline{5},\, \underline{12},\, \underline{19},\, \underline{26},\, 33\)
\(\,\,\,\,\,\,\,\,\,-9,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 33\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,33=-9+d(7-1)\)
\(\,\,\,\,\,\,\,\,\,33=-9+6d\)
\(\,\,\,\,\,\,\,\,\,42=6d\)
\(\,\,\,\,\,\,\,\,\,7=d\)
\(\,\,\,\,\,\,\,\,\,-9,\, \underline{-2},\, \underline{5},\, \underline{12},\, \underline{19},\, \underline{26},\, 33\)
\(\textbf{17)}\) Find the four arithmetic means of \(5\) and \(35\)\(\,\,\,\)The answer is \(11,17,23,29\)
\(\,\,\,\,\,\,\,\,\,5,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 35\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,35=5+d(6-1)\)
\(\,\,\,\,\,\,\,\,\,35=5+5d\)
\(\,\,\,\,\,\,\,\,\,30=5d\)
\(\,\,\,\,\,\,\,\,\,6=d\)
\(\,\,\,\,\,\,\,\,\,5,\, \underline{11},\, \underline{17},\, \underline{23},\, \underline{29},\, 35\)
\(\,\,\,\,\,\,\,\,\,5,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 35\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,35=5+d(6-1)\)
\(\,\,\,\,\,\,\,\,\,35=5+5d\)
\(\,\,\,\,\,\,\,\,\,30=5d\)
\(\,\,\,\,\,\,\,\,\,6=d\)
\(\,\,\,\,\,\,\,\,\,5,\, \underline{11},\, \underline{17},\, \underline{23},\, \underline{29},\, 35\)
\(\textbf{18)}\) Find the two arithmetic means of \(-1.5\) and \(7.5\)\(\,\,\,\)The answer is \(1.5,4.5\)
\(\,\,\,\,\,\,\,\,\,-1.5,\, \text{___},\, \text{___},\, 7.5\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,7.5=-1.5+d(4-1)\)
\(\,\,\,\,\,\,\,\,\,7.5=-1.5+3d\)
\(\,\,\,\,\,\,\,\,\,9=3d\)
\(\,\,\,\,\,\,\,\,\,3=d\)
\(\,\,\,\,\,\,\,\,\,-1.5,\, \underline{-1.5+3},\, \underline{-1.5+6},\, 7.5\)
\(\,\,\,\,\,\,\,\,\,-1.5,\, \underline{1.5},\, \underline{4.5},\, 7.5\)
\(\,\,\,\,\,\,\,\,\,-1.5,\, \text{___},\, \text{___},\, 7.5\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,7.5=-1.5+d(4-1)\)
\(\,\,\,\,\,\,\,\,\,7.5=-1.5+3d\)
\(\,\,\,\,\,\,\,\,\,9=3d\)
\(\,\,\,\,\,\,\,\,\,3=d\)
\(\,\,\,\,\,\,\,\,\,-1.5,\, \underline{-1.5+3},\, \underline{-1.5+6},\, 7.5\)
\(\,\,\,\,\,\,\,\,\,-1.5,\, \underline{1.5},\, \underline{4.5},\, 7.5\)
\(\textbf{19)}\) Find the three arithmetic means of \(0\) and \(100\)\(\,\,\,\)The answer is \(25,50,75\)
\(\,\,\,\,\,\,\,\,\,0,\, \text{___},\, \text{___},\, \text{___},\, 100\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,100=0+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,100=4d\)
\(\,\,\,\,\,\,\,\,\,25=d\)
\(\,\,\,\,\,\,\,\,\,0,\, \underline{25},\, \underline{50},\, \underline{75},\, 100\)
\(\,\,\,\,\,\,\,\,\,0,\, \text{___},\, \text{___},\, \text{___},\, 100\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,100=0+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,100=4d\)
\(\,\,\,\,\,\,\,\,\,25=d\)
\(\,\,\,\,\,\,\,\,\,0,\, \underline{25},\, \underline{50},\, \underline{75},\, 100\)
\(\textbf{20)}\) How many arithmetic means are inserted between \(4\) and \(34\) if the common difference is \(5\)?\(\,\,\,\)The answer is \(5\) arithmetic means
\(\,\,\,\,\,\,\,\,\,4,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 34\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,34=4+5(n-1)\)
\(\,\,\,\,\,\,\,\,\,30=5(n-1)\)
\(\,\,\,\,\,\,\,\,\,6=n-1\)
\(\,\,\,\,\,\,\,\,\,7=n\)
\(\,\,\,\,\,\,\,\,\,\text{There are }7\text{ total terms.}\)
\(\,\,\,\,\,\,\,\,\,7-2=5\text{ arithmetic means}\)
\(\,\,\,\,\,\,\,\,\,4,\,9,\,14,\,19,\,24,\,29,\,34\)
\(\,\,\,\,\,\,\,\,\,4,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 34\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,34=4+5(n-1)\)
\(\,\,\,\,\,\,\,\,\,30=5(n-1)\)
\(\,\,\,\,\,\,\,\,\,6=n-1\)
\(\,\,\,\,\,\,\,\,\,7=n\)
\(\,\,\,\,\,\,\,\,\,\text{There are }7\text{ total terms.}\)
\(\,\,\,\,\,\,\,\,\,7-2=5\text{ arithmetic means}\)
\(\,\,\,\,\,\,\,\,\,4,\,9,\,14,\,19,\,24,\,29,\,34\)
See Related Pages
\(\bullet\text{ Arithmetic Sequences}\)
\(\,\,\,\,\,\,\,a_n=a_1 + d(n-1)\)
\(\bullet\text{ Geometric Sequences}\)
\(\,\,\,\,\,\,\,a_n=a_1 \cdot r^{(n-1)}…\)
\(\bullet\text{ Arithmetic Series}\)
\(\,\,\,\,\,\,\,s_n=\frac{n}{2}(a_1+a_n)…\)
\(\bullet\text{ Geometric Series}\)
\(\,\,\,\,\,\,\,s_n=a_1 \frac{1-r^n}{1-r}…\)
\(\bullet\text{ Infinite Geometric Series}\)
\(\,\,\,\,\,\,\,s_\infty = \frac{a_1}{1-r}\,\,\, |r| \lt 1…\)
\(\bullet\text{ Recursive Sequences}\)
\(\,\,\,\,\,\,\, a_{1}=2, \, a_{n+1}=a_{n}+3…\)