\(\textbf{1)}\) Find the three arithmetic means of \(3\) and \(48\)
The answer is \(6,12,24\)
\(\textbf{2)}\) Find the four arithmetic means of \(56\) and \(356\)
\(\,\,\,\)The answer is \(116,176,236,296\)
\(\,\,\,\,\,\,\,\,\,56,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 356\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,356=56+d(6-1)\)
\(\,\,\,\,\,\,\,\,\,356=56+d(5)\)
\(\,\,\,\,\,\,\,\,\,300=d(5)\)
\(\,\,\,\,\,\,\,\,\,60=d\)
\(\,\,\,\,\,\,\,\,\,56,\, \underline{56+60},\, \underline{56+120},\, \underline{56+180},\, \underline{56+240},\, 356\)
\(\,\,\,\,\,\,\,\,\,56,\, \underline{116},\, \underline{176},\, \underline{236},\, \underline{296},\, 356\)
\(\textbf{3)}\) Find the three arithmetic means of \(2\) and \(10\)
\(\,\,\,\)The answer is \(4,6,8\)
\(\,\,\,\,\,\,\,\,\,2,\, \text{___},\, \text{___},\, \text{___},\, 10\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,10=2+d(5-1)\)
\(\,\,\,\,\,\,\,\,\,10=2+d(4)\)
\(\,\,\,\,\,\,\,\,\,8=d(4)\)
\(\,\,\,\,\,\,\,\,\,2=d\)
\(\,\,\,\,\,\,\,\,\,2,\, \underline{2+2},\, \underline{2+4},\, \underline{2+6},\, 10\)
\(\,\,\,\,\,\,\,\,\,2,\, \underline{4},\, \underline{6},\, \underline{8},\, 10\)
\(\textbf{4)}\) Find the two arithmetic means of \(-5\) and \(40\)
\(\,\,\,\)The answer is \(10,25\)
\(\,\,\,\,\,\,\,\,\,-5,\, \text{___},\, \text{___},\, 55\)
\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)
\(\,\,\,\,\,\,\,\,\,40=-5+d(4-1)\)
\(\,\,\,\,\,\,\,\,\,40=-5+d(3)\)
\(\,\,\,\,\,\,\,\,\,45=d(3)\)
\(\,\,\,\,\,\,\,\,\,15=d\)
\(\,\,\,\,\,\,\,\,\,-5,\, \underline{-5+15},\, \underline{-5+30},\, 40\)
\(\,\,\,\,\,\,\,\,\,-5,\, \underline{10},\, \underline{25},\, 40\)
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