Arithmetic Means Between 2 Terms

\(\textbf{1)}\) Find the three arithmetic means of \(3\) and \(48\)

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The answer is \(6,12,24\)

\(\textbf{2)}\) Find the four arithmetic means of \(56\) and \(356\)

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\(\,\,\,\)The answer is \(116,176,236,296\)

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\(\,\,\,\,\,\,\,\,\,56,\, \text{___},\, \text{___},\, \text{___},\, \text{___},\, 356\)

\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)

\(\,\,\,\,\,\,\,\,\,356=56+d(6-1)\)

\(\,\,\,\,\,\,\,\,\,356=56+d(5)\)

\(\,\,\,\,\,\,\,\,\,300=d(5)\)

\(\,\,\,\,\,\,\,\,\,60=d\)

\(\,\,\,\,\,\,\,\,\,56,\, \underline{56+60},\, \underline{56+120},\, \underline{56+180},\, \underline{56+240},\, 356\)

\(\,\,\,\,\,\,\,\,\,56,\, \underline{116},\, \underline{176},\, \underline{236},\, \underline{296},\, 356\)

\(\textbf{3)}\) Find the three arithmetic means of \(2\) and \(10\)

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\(\,\,\,\)The answer is \(4,6,8\)

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\(\,\,\,\,\,\,\,\,\,2,\, \text{___},\, \text{___},\, \text{___},\, 10\)

\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)

\(\,\,\,\,\,\,\,\,\,10=2+d(5-1)\)

\(\,\,\,\,\,\,\,\,\,10=2+d(4)\)

\(\,\,\,\,\,\,\,\,\,8=d(4)\)

\(\,\,\,\,\,\,\,\,\,2=d\)

\(\,\,\,\,\,\,\,\,\,2,\, \underline{2+2},\, \underline{2+4},\, \underline{2+6},\, 10\)

\(\,\,\,\,\,\,\,\,\,2,\, \underline{4},\, \underline{6},\, \underline{8},\, 10\)

\(\textbf{4)}\) Find the two arithmetic means of \(-5\) and \(40\)

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\(\,\,\,\)The answer is \(10,25\)

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\(\,\,\,\,\,\,\,\,\,-5,\, \text{___},\, \text{___},\, 55\)

\(\,\,\,\,\,\,\,\,\,a_n=a_1+d(n-1)\)

\(\,\,\,\,\,\,\,\,\,40=-5+d(4-1)\)

\(\,\,\,\,\,\,\,\,\,40=-5+d(3)\)

\(\,\,\,\,\,\,\,\,\,45=d(3)\)

\(\,\,\,\,\,\,\,\,\,15=d\)

\(\,\,\,\,\,\,\,\,\,-5,\, \underline{-5+15},\, \underline{-5+30},\, 40\)

\(\,\,\,\,\,\,\,\,\,-5,\, \underline{10},\, \underline{25},\, 40\)

See Related Pages

\(\bullet\text{ Arithmetic Sequences}\)
\(\,\,\,\,\,\,\,a_n=a_1 + d(n-1)\)

\(\bullet\text{ Geometric Sequences}\)
\(\,\,\,\,\,\,\,a_n=a_1 \cdot r^{(n-1)}…\)

\(\bullet\text{ Arithmetic Series}\)
\(\,\,\,\,\,\,\,s_n=\frac{n}{2}(a_1+a_n)…\)

\(\bullet\text{ Geometric Series}\)
\(\,\,\,\,\,\,\,s_n=a_1 \frac{1-r^n}{1-r}…\)

\(\bullet\text{ Infinite Geometric Series}\)
\(\,\,\,\,\,\,\,s_\infty = \frac{a_1}{1-r}\,\,\, |r| \lt 1…\)

\(\bullet\text{ Recursive Sequences}\)
\(\,\,\,\,\,\,\, a_{1}=2, \, a_{n+1}=a_{n}+3…\)

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