Question

Three rational numbers A, B, and C are inserted between −1/6 and 2/3 such that the five numbers are equally spaced. Find the values of A, B, and C. Express your answers as fractions in their simplest forms.

Answer 1

A fraction is in simplest form when the top and bottom cannot be any smaller, while still being whole numbers. To simplify a fraction: divide the top and bottom by the greatest number that will divide both numbers exactly (they must stay whole numbers)...~~~

Answer 2

Basic Concept Used :-

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

Let's solve the problem now!!!

$\purple{\large\underline{\bf{Solution-}}}$

Since, it is given that A, B and C are inserted between two rational numbers so that they are equally spaced.

Tʜᴜs,

$\rm :\longmapsto\: - \dfrac{1}{6},A, B,C,\dfrac{2}{3} \: forms \: an \: AP \: series$

where,

$\red{\rm :\longmapsto\:a_1 = - \dfrac{1}{6}} \\ \red{\rm :\longmapsto\:a_n = \dfrac{2}{3}} \: \: \: \\ \red{\rm :\longmapsto\:n =5 \: \: \: \: \: }$

So, using formula,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

On substituting the values, we get

$\rm :\longmapsto\:\dfrac{2}{3} = - \dfrac{1}{6} + (5 - 1)d$

$\rm :\longmapsto\:\dfrac{2}{3} + \dfrac{1}{6} = 4d$

$\rm :\longmapsto\:\dfrac{4 + 1}{6} = 4d$

$\rm :\longmapsto\:\dfrac{5}{6} = 4d$

$\bf\implies \:d = \dfrac{5}{24}$

Thus,

$\rm :\longmapsto\:A = a_2 = a + d$

$\rm = \: \: - \dfrac{1}{3} + \dfrac{5}{24}$

$\rm = \: \: \dfrac{ - 8 + 5}{24}$

$\rm = \: \: \dfrac{ - 3}{24}$

$\rm = \: \: \dfrac{ - 1}{8}$

$\bf\implies \:A = - \dfrac{1}{8}$

$\rm :\longmapsto\:B = a_3 = a + 2d$

$\rm = \: \: - \dfrac{1}{3} +2 \times \dfrac{5}{24}$

$\rm = \: \: - \dfrac{1}{3} + \dfrac{5}{12}$

$\rm = \: \: \dfrac{ - 4 + 5}{12}$

$\rm = \: \: \dfrac{ 1}{12}$

$\bf\implies \:B = \dfrac{1}{12}$

$\rm :\longmapsto\:C = a_4 = a + 3d$

$\rm = \: \: - \dfrac{1}{3} +3 \times \dfrac{5}{24}$

$\rm = \: \: - \dfrac{1}{3} + \dfrac{5}{8}$

$\rm = \: \: \dfrac{ - 8 + 15}{24}$

$\rm = \: \: \dfrac{7}{24}$

$\bf\implies \:C = \dfrac{7}{24}$

Additional Information :-

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of first n terms of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(2\:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

Or

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(\:a\:+\:a_n \bigg)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.